A Song of Ice and Fire
We know that Dany was born to King Aerys Targaryen II and Queen Rhaella Targaryen, we also know that Rhaella was sent to Dragonstone after the Battle of the Trident and before the Sack of King's Landing, so sometime in 283 AC.
"Rhaegar met Robert on the Trident, and you know what happened there. When the word reached court, Aerys packed the queen off to Dragonstone with Prince Viserys. Princess Elia would have gone as well, but he forbade it. Somehow he had gotten it in his head that Prince Lewyn must have betrayed Rhaegar on the Trident, but he thought he could keep Dorne loyal so long as he kept Elia and Aegon by his side.
A Storm of Swords, Jaime V
And if it isn't obvious from the situation that Rhaella must have already been pregnant The World of Ice and Fire spells it out for us.
Birds flew and couriers raced to bear word of the victory at the Ruby Ford. When the news reached the Red Keep, it was said that Aerys cursed the Dornish, certain that Lewyn had betrayed Rhaegar. He sent his pregnant queen, Rhaella, and his younger son and new heir, Viserys, away to Dragonstone, but Princess Elia was forced to remain in King's Landing with Rhaegar's children as a hostage against Dorne.
The World of Ice and Fire, The Fall of the Dragons: The End
Dany herself mentions she was born "nine moons" after the flight to Dragonstone placing her birth at the end of 283 AC or in 284 AC, however, it is more likely to be 284 AC.
She had been born on Dragonstone nine moons after their flight, while a raging summer storm threatened to rip the island fastness apart. They said that storm was terrible. The Targaryen fleet was smashed while it lay at anchor, and huge stone blocks were ripped from the parapets and sent hurtling into the wild waters of the narrow sea. Her mother had died birthing her, and for that her brother Viserys had never forgiven her.
A Game of Thrones, Daenerys I
Eddard Stark mentions on a couple of occassions about 14 years having passed from King Robert Baratheon being crowned but he explicitly states Dany's age twice and this happens in 298 AC as we know the series starts in this year.
"Whereas Daenerys is a fourteen-year-old girl." Ned knew he was pushing this well past the point of wisdom, yet he could not keep silent. "Robert, I ask you, what did we rise against Aerys Targaryen for, if not to put an end to the murder of children?"
A Game of Thrones, Eddard VIII
Ned had heard enough. "You send hired knives to kill a fourteen-year-old girl and still quibble about honor?" He pushed back his chair and stood. "Do it yourself, Robert. The man who passes the sentence should swing the sword. Look her in the eyes before you kill her. See her tears, hear her last words. You owe her that much at least."
A Game of Thrones, Eddard VIII
This still leaves both years as being possible for Dany to have been born in. However, we learn that Dany finds out she is pregnant on her 14th name day.
They were on the far side of the Dothraki sea when Jhiqui brushed the soft swell of Dany's stomach with her fingers and said, "Khaleesi, you are with child."
"I know," Dany told her.
It was her fourteenth name day.
A Game of Thrones, Daenerys III
Her pregnancy is discussed in Kings Landing sometime after the Hand's Tourney.
"The whore is pregnant!" The king's fist slammed down on the council table loud as a thunderclap. "I warned you this would happen, Ned. Back in the barrowlands, I warned you, but you did not care to hear it. Well, you'll hear it now. I want them dead, mother and child both, and that fool Viserys as well. Is that plain enough for you? I want them dead."
A Game of Thrones, Eddard VIII
The following article mentions that the Hand's Tourney takes place in 298 AC though I don't know if this is referenced elsewhere. As we know the series starts in 298 AC this would mean Dany more than likely turns 14 in 298 AC placing her birth in 284 AC.
However, for something more concrete Dany mentions she is still 14 when she is following the comet.
"We follow the comet," Dany told her khalasar. Once it was said, no word was raised against it. They had been Drogo's people, but they were hers now. The Unburnt, they called her, and Mother of Dragons. Her word was their law.
They are not strong, she told herself, so I must be their strength. I must show no fear, no weakness, no doubt. However frightened my heart, when they look upon my face they must see only Drogo's queen. She felt older than her fourteen years. If ever she had truly been a girl, that time was done.
A Clash of Kings, Daenerys I
The following wiki page asserts that the comet appears in the sky in 299 AC.
The first time the Red Comet is mentioned to have been seen, occurs after Rickon Stark has turned 4. Thus all POVs where the Red Comet appears can be said to take place in 299 AC with certainty.
Years after Aegon's Conquest/Calculations Events
Note that Rickon is born in the year 295 AC and though I won't go adding quotes here, here's the summary of the calculation.
Rickon was 3 years old when the direwolves are found and during the welcoming feast at Winterfell. Both events occur in 298 AC, giving the following options:
2 turning 3 in 298 AC, 3 turning 4 in 299 AC, 4 turning 5 in 300 AC, placing his birth in 295 AC
3 turning 4 in 298 AC, 4 turning 5 in 299 AC, 5 turning 6 in 300 AC, placing his birth in 294 AC
Rickon was “near four” when Robb left for war, and he was 4 when the news of Ned's death reached Winterfell. Rickon was still four when Theon Greyjoy attacked Winterfell. After the Sack of Winterfell Rickon is still at 4. In 300 AC, Rickon is still 4 years old. This shows that the only correct option for Rickon is 3 turning 4 in 299 AC, placing his birth in 295 AC.
Years after Aegon's Conquest/Calculations Ages (Continued3)
As Dany has now been mentioned to be 14 in both 298 AC and 299 AC she must have been born in 284 AC.
Game of Thrones
Side note: All years on the wikia are given in AL instead of AC. Since answering this question I asked the following about the change. It turns out that the show committed to AL before AC was confirmed by GRRM as AL was the popular year convention by fans.
Her birth is stated on her wikia page and is cited as shortly after Robert's Rebellion. Robert's rebellion took place around 280 AL to 282 AL according to the page and so she must have been born in 282 AL or 283 AL. However it is stated that the rebelleion ended on Dragonstone in 282 AL and as Dany had been born there and fled before the end she must have been born in 282 AL. This also places her age as 16 turning 17 in 298 AL, the start of the show.
The following westeros.org page seems to contradict this saying the following, though I don't know where it is sourced from.
Daenerys has been aged for the show. In the novels, she is 13 when she is first introduced. In Game of Thrones, she is 15 years old. Contacts to give her violet eyes were tried in the pilot, but proved more trouble than they were worth.
Westeros, Daenerys Targaryen
It appears as though they were keeping the birth year the same as the books when the looked into their calculations. However, the following article backs up the wikia stating that Dany is 17 at the start of the show which reinforces her birth in 282 AL, where it is sourced from I don't know.
Age in the books: In A Game of Thrones (the first book in George RR Martin’s A Song of Ice and Fire series), Daenerys is around 13 years old.
Age in the TV show: In HBO’s adaptation, it is stated in season one that 17 years have passed since Robert Baratheon’s rebellion – around the time Daenerys was born. It is therefore reasonable to suggest that Daenerys starts the show as a 17 year old. By season six, around five years have passed since the first episode, so we can safely assume that Daenerys is in her early twenties, let’s say, 22 years old. It’s easy to understand why they increased Dany’s age for the show since she had to shoot some pretty uncomfortable and risqué scenes with Khal Drogo from the start of the series.
Game of Thrones ages: just how old are Jon, Sansa, Brienne and Daenerys meant to be?
GRRM has even gone on record talking about why the show changed the ages and mentions the age increase for the show.
There was a fair amount of explicit sex in the series and some fans of the books were taken aback.
One of the reasons I wanted to do this with HBO is that I wanted to keep the sex. We had some real problems because Dany is only 13 in the books, and that’s based on medieval history. They didn’t have this concept of adolescence or the teenage years. You were a child or you were an adult. And the onset of sexual maturity meant you were an adult. So I reflected that in the books. But then when you go to film it you run into people going crazy about child pornography and there’s actual laws about how you can’t depict a 13 year old having sex even if you have an 18 year old acting the part — it’s illegal in the United Kingdom. So we ended up with a 22 year old portraying an 18 year old, instead of an 18 year old portraying a 13 year old. If we decided to lose the sex we could have kept the original ages. And once you change the age of one character you have to change the ages of all the characters, and change the date of the war [that dethroned the Mad King]. The fact we made all these changes indicates how important we thought sex was.
Entertainment Weekly, A Dance With Dragons Interview
Note he states her age as 18 at the start of the show but that could be an interview transcript typo, a mistake or it was changed to 17 at a later date.