Is it possible that during Hagrid's time in Azkaban he talked with Sirius and told him of Harry being alive, and that's why Sirius escapes?
Sirius gives his reasons for escaping - and those reasons had everything to do with Pettigrew being alive, while Sirius was stuck in Azkaban for Pettigrew's crimes. This was found out from Fudge's newspaper, not Hagrid.
"You know, Sirius, that's a fair question," said Lupin, turning to Black and frowning slightly. "How did you find out where he was?"
Black put one of his clawlike hands inside his robes and took out a crumpled piece of paper, which he smoothed flat and held out to show the others.
It was the photograph of Ron and his family that had appeared in the Daily Prophet the previous summer, and there, on Ron's shoulder, was Scabbers.
"How did you get this?" Lupin asked Black, thunderstruck.
"Fudge," said Black. "When he came to inspect Azkaban last year, he gave me his paper. And there was Peter, on the front page... on this boy's shoulder... I knew him at once... how many times had I seen him transform? And the caption said the boy would be going back to Hogwarts... to where Harry was..."
Harry Potter and the Prisoner of Azkaban, chapter 19
Sirius has no reason to presume Harry was ever dead. Surviving Voldemort was a well known event. Harry is still not dead, nor was ever assumed to be anything other than not dead, up until Prisoner of Azkaban, when Sirius escapes. Sirius knew quite well Harry was alive - he offered to take him to live with him.
"How was I ter know he wasn' upset abou' Lily an' James? It was You-Know-Who he cared abou'! An' then he says, 'Give Harry ter me, Hagrid, I'm his godfather, I'll look after him -' Ha! But I'd had me orders from Dumbledore, an' I told Black no, Dumbledore said Harry was ter go ter his aunt an' uncle's. Black argued, but in the end he gave in. Told me ter take his motorbike ter get Harry there. 'I won't need it anymore,' he says.
"I shoulda known there was somethin' fishy goin' on then. He loved that motorbike, what was he givin' it ter me for? Why wouldn' he need it anymore? Fact was, it was too easy ter trace. Dumbledore knew he'd bin the Potters' Secret-Keeper. Black knew he was goin' ter have to run for it that night, knew it was a matter of time before the Ministry was after him.
"But what if I'd given Harry to him, eh? I bet he'd've pitched him off the bike halfway out ter sea. His bes' friends' son! But when a wizard goes over ter the Dark Side, there's nothin' and no one that matters to 'em anymore..."
Harry Potter and the Prisoner of Azkaban, chapter 10
Hagrid would have very little to say to any Azkaban prisoner, unless they wanted to talk dragons. Sirius being known as a mass-murderer and thinking he may be involved with the Potters' deaths by Voldemort, he'd have even less to say.
"An' them dementors make me feel ruddy terrible an' all," said Hagrid, with a shudder. "Gotta walk past 'em ev'ry time I want a drink in The Three Broomsticks. 'S like bein' back in Azkaban -"
He fell silent, gulping his tea. Harry, Ron, and Hermione watched him breathlessly. They had never heard Hagrid talk about his brief spell in Azkaban before. After a pause, Hermione said timidly, "Is it awful in there, Hagrid?"
"Yeh've no idea," said Hagrid quietly. "Never bin anywhere like it. Thought I was goin' mad. Kep' goin' over horrible stuff in me mind... the day I got expelled from Hogwarts.. . day me dad died... day I had ter let Norbert go..."
His eyes filled with tears. Norbert was the baby dragon Hagrid had once won in a game of cards.
"Yeh can' really remember who yeh are after a while. An' yeh can' see the point o' livin' at all. I used ter hope I'd jus' die in me sleep... When they let me out, it was like bein' born again, ev'rythin' came floodin' back, it was the bes' feelin' in the world. Mind, the dementors weren't keen on lettin' me go."
Harry Potter and the Prisoner of Azkaban, chapter 11. (See also in chapter 10, at Hagrid's speech about Sirius showing up at the Potter's house after the attack, in The Three Broomsticks.)
Fudge's descriptions of Azkaban indicates he's rarely if ever out of his cell for random socializing with other criminals. Fudge came to observe him directly in his room; other than escaping he never mentions being elsewhere on the island.
Madam Rosmerta let out a long sigh.
"Is it true he's mad, Minister?"
"I wish I could say that he was," said Fudge slowly. "I certainly believe his master's defeat unhinged him for a while. The murder of Pettigrew and all those Muggles was the action of a cornered and desperate man - cruel... pointless. Yet I met Black on my last inspection of Azkaban. You know, most of the prisoners in there sit muttering to themselves in the dark; there's no sense in them... but I was shocked at how normal Black seemed. He spoke quite rationally to me. It was unnerving. You'd have thought he was merely bored - asked if I'd finished with my newspaper, cool as you please, said he missed doing the crossword. Yes, I was astounded at how little effect the dementors seemed to be having on him - and he was one of the most heavily guarded in the place, you know. Dementors outside his door day and night."
Harry Potter and the Prisoner of Azkaban, chapter 10