Yes Daenerys did know what would happen due to her intelligence. It is very heavily implied.
But she was not entirely sure. It was a leap of faith, she knew the risks and she decided to take them.
Signs before the act
Daenerys observed many curious things before she came to the conclusion that the eggs could be hatched.
Daenerys had her first dragon dream before she was even given the eggs:
Yet that night she dreamt of one. Viserys was hitting her, hurting
her. She was naked, clumsy with fear. She ran from him, but her body
seemed thick and ungainly. He struck her again. She stumbled and fell.
"You woke the dragon," he screamed as he kicked her. "You woke the
dragon, you woke the dragon." Her thighs were slick with blood. She
closed her eyes and whimpered. As if in answer, there was a hideous
ripping sound and the crackling of some great fire. When she looked
again, Viserys was gone, great columns of flame rose all around, and
in the midst of them was the dragon. It turned its great head slowly. When its molten eyes found hers, she woke, shaking and covered
with a fine sheen of sweat. She had never been so afraid
…
AGOT - Daenerys II
First thing she saw was red flames when light touched her eggs.
Dany gave the silver over to the slaves for grooming and entered her
tent. It was cool and dim beneath the silk. As she let the door flap
close behind her, Dany saw a finger of dusty red light reach out to
touch her dragon's eggs across the tent. For an instant a thousand
droplets of scarlet flame swam before her eyes. She blinked, and
they were gone.
AGOT - Daenerys III
Then she observed that when she touched the eggs, they felt hot. But she told herself that it was just because of sunlight.
Stone, she told herself. They are only stone, even Illyrio said so,
the dragons are all dead. She put her palm against the black egg,
fingers spread gently across the curve of the shell. The stone was
warm. Almost hot. "The sun," Dany whispered. "The sun warmed them as
they rode."
AGOT - Daenerys III
Yet the Eggs kept feeling warm on her touch.
“Khaleesi, “ Jhiqui said, “what is wrong? Are you sick?”
“I was,” she answered, standing over the dragon’s eggs that Illyrio
had given her when she wed. She touched one, the largest of the
three, running her hand lightly over the shelf. Black-and- scarlet,
she thought, like the dragon in my dream. The stone felt strangely
warm beneath her fingers... or was she still dreaming? She pulled
her hand back nervously.
AGOT - Daenerys III
The curious thing was, only she could feel the heat, no one else could.
She saw a dragon dream again where she was doused with dragonfire and emerged unscathed, stronger and fiercer.
she dreamt the dragon dream again. Viserys was not in it this time. There was only her and the dragon. Its scales were black as
night, wet and slick with blood. Her blood, Dany sensed. Its eyes were
pools of molten magma, and when it opened its mouth, the flame came
roaring out in a hot jet. She could hear it singing to her. She
opened her arms to the fire, embraced it, let it swallow her whole,
let it cleanse her and temper her and scour her clean. She could
feel her flesh sear and blacken and slough away, could feel her blood
boil and turn to steam, and yet there was no pain. She felt strong
and new and fierce.
AGOT - Daenerys III
Then she felt her unborn son reaching out to dragon-egg:
She was lying there, holding the egg, when she felt the child move
within her... as if he were reaching out, brother to brother, blood to
blood. “You are the dragon,” Dany whispered to him, “the true dragon.
I know it. I know it.”
AGOT - Daenerys IV
When Robert's catspaw tried to kill her, she was very much shaken. She then tried to hatch the dragons, but failed.
When the coals were afire, Dany sent Ser Jorah from her.
She had to be alone to do what she must do. This is madness, she told
herself as she lifted the black-and-scarlet egg from the velvet. It
will only crack and burn, and it's so beautiful, Ser Jorah will call
me a fool if I ruin it, and yet, and yet …
Cradling the egg with both hands, she carried it to the fire and
pushed it down amongst the burning coals. The black scales seemed to
glow as they drank the heat. Flames licked against the stone with
small red tongues. Dany placed the other two eggs beside the black one
in the fire. As she stepped back from the brazier, the breath trembled
in her throat. She watched until the coals had turned to ashes. Drifting sparks floated up and out of the smokehole. Heat shimmered in waves around the dragon's eggs. And that was all.
AGOT - Daenerys VI
How did it happen?
She first came to know about how the blood magic worked when the Maegi gave away too much.
she pleaded. “Save him, and I will free you, I swear it. You must know
a way... some magic, some...”
Mirri Maz Duur sat back on her heels and studied Daenerys through eyes
as black as night. “There is a spell.” Her voice was quiet, scarcely
more than a whisper. “But it is hard, lady, and dark. Some would say
that death is cleaner. I learned the way in Asshai, and paid dear for
the lesson. My teacher was a bloodmage from the Shadow Lands.”
Dany went cold all over. “Then you truly are a maegi.”
“Am I? Mirri Maz Duur smiled. “Only a maegi can save your rider now,
Silver Lady.”
“Is there no other way?”
“No other.”
Khal Drogo gave a shuddering gasp.
“Do it,” Dany blurted. She must not be afraid; she was the blood of
the dragon. “Save him.”
“There is a price,” the godswife warned her.
“You’ll have gold, horses, whatever you like.”
“It is not a matter of gold or horses. This is bloodmagic, lady. Only
death may pay for life.”
AGOT - Daenerys VIII
This was later confirmed by the Maegi later:
You warned me that only death could pay for life. I thought you meant
the horse.”
“No,” Mirri Maz Duur said. “That was a lie you told yourself. You knew
the price.”
Had she? Had she? If I look back I am lost. “The price was paid,” Dany
said.
AGOT - Daenerys IX
Daenerys now had a fair idea of how exactly does the magic work. She was about to test it on Drogo's funeral pyre. The Maegi however tried to bluff that the sacrifice alone is not enough.
“It is not enough to kill a horse,” she told Dany. “By itself, the
blood is nothing. You do not have the words to make a spell, nor the
wisdom to find them. Do you think bloodmagic is a game for children?
You call me maegi as if it were a curse, but all it means is wise. You
are a child, with a child’s ignorance. Whatever you mean to do, it
will not work. Loose me from these bonds and I will help you.”
AGOT - Daenerys X
Daenerys however learnt not to be bitten by the same snake twice. She didn't take the bait. People around her thought that she was either trying to resurrect Drogo or worst, trying to commit suicide out of grief.
But Daenerys clearly pointed out that she was not committing suicide. This implies she knew what she was doing and she knew what would be the result.
“I understand that you loved him,” Ser Jorah said in a voice thick
with despair. “I loved my lady wife once, yet I did not die with her.
You are my queen, my sword is yours, but do not ask me to stand aside
as you climb on Drogo’s pyre. I will not watch you burn.”
“Is that what you fear?” Dany kissed him lightly on his broad
forehead. “I am not such a child as that, sweet ser.”
“You do not mean to die with him? You swear it, my queen?”
“I swear it,”
AGOT - Daenerys X
She then proceeded with claiming her husband's khalasar's remnants as her own and demanding oaths of fealty from her retainers. These are clearly not the acts of someone who was about to embark on an unknown journey or to commit suicide. She clearly had a plan for the future.
Daenerys was however not entirely sure. She was gambling and she knew it.
They thought her mad, Dany realized. Perhaps she was. She would know
soon enough. If I look back I am lost.
AGOT - Daenerys X
She then asked for forgiveness from Drogo, as she was about to use him as sacrifice for one of her eggs.
She buried her face in it and inhaled the dark fragrance of the oils.
He smelled like grass and warm earth, like smoke and semen and horses.
He smelled like Drogo. Forgive me, sun of my life, she thought.
Forgive me for all I have done and all I must do. I paid the price, my
star, but it was too high, too high...
AGOT - Daenerys X
Jorah tried to persuade her to sell the eggs, Daenerys however flat-out refused, implicitly claiming that the eggs were given to her for a nobler purpose than accumulating wealth:
My queen, Drogo will have no use for dragon’s eggs in the night lands.
Better to sell them in Asshai. Sell one and we can buy a ship to take
us back to the Free Cities. Sell all three and you will be a wealthy
woman all your days.”
“They were not given to me to sell,” Dany told him.
AGOT - Daenerys X
Daenerys then implicitly told the Maegi what she was about to do:
“You will not hear me scream,” Mirri responded as the oil dripped from
her hair and soaked her clothing.
“I will,” Dany said, “but it is not your screams I want, only your
life. I remember what you told me. Only death can pay for life.” Mirri
Maz Duur opened her mouth, but made no reply. As she stepped away,
Dany saw that the contempt was gone from the maegi’s flat black eyes;
in its place was something that might have been fear. Then there was
nothing to be done but watch the sun and look for the first star.
AGOT - Daenerys X
The star she was looking for was actually a sign for her.
Jhogo spied it first. “There, “ he said in a hushed voice. Dany looked
and saw it, low in the east. The first star was a comet, burning red.
Bloodred; fire red; the dragon’s tail. She could not have asked for a
stronger sign.
AGOT - Daenerys X
The following passage however clearly states that Daenerys knew dragons would hatch with fire but she did not know that she had to offer sacrifices as well.
She had sensed the truth of it long ago, Dany thought as she took a
step closer to the conflagration, but the brazier had not been hot
enough.
AGOT - Daenerys X
Even before she saw her dragons, she knew her plan had worked.
And something else came crashing down, bouncing and rolling, to land
at her feet; a chunk of curved rock, pale and veined with gold, broken
and smoking. The roaring filled the world, yet dimly through the
firefall Dany heard women shriek and children cry out in wonder.
Only death can pay for life.
And there came a second crack, loud and sharp as thunder, and the
smoke stirred and whirled around her and the pyre shifted, the logs
exploding as the fire touched their secret hearts. She heard the
screams of frightened horses, and the voices of the Dothraki raised in
shouts of fear and terror, and Ser Jorah calling her name and cursing.
No, she wanted to shout to him, no, my good knight, do not fear.for
me. The fire is mine. I am Daenerys Stormborn, daughter of dragons,
bride of dragons, mother of dragons, don’t you see? Don’t you SEE?
AGOT - Daenerys X
So yes, she did know that this was gonna hatch dragon-eggs.
Conclusion
- Daenerys knew that fire was needed to hatch the dragons. She however did not know that blood was needed as well.
- Daenerys had a fair idea that dragons would hatch however she was not sure. This was a risk she took.
- Daenerys was acting on a plan and it worked out. At no point did she show any intentions of committing suicide. She was planning for the future all along.
