With everything that has happened in the show, I was just wondering how much Daenerys and Jon know about eachother in the books, if anything?
Daenerys almost certainly has no idea Jon exists. Hell, her knowledge of Eddard Stark is scant at best:
Yet sometimes Dany would picture the way it had been, so often had her brother told her the stories. The midnight flight to Dragonstone, moonlight shimmering on the ship’s black sails. Her brother Rhaegar battling the Usurper in the bloody waters of the Trident and dying for the woman he loved. The sack of King’s Landing by the ones Viserys called the Usurper’s dogs, the lords Lannister and Stark.
The Usurper will kill you, sure as sunrise, Mormont had said. Robert had slain her gallant brother Rhaegar, and one of his creatures had crossed the Dothraki sea to poison her and her unborn son. They said Robert Baratheon was strong as a bull and fearless in battle, a man who loved nothing better than war. And with him stood the great lords her brother had named the Usurper’s dogs, cold-eyed Eddard Stark with his frozen heart, and the golden Lannisters, father and son, so rich, so powerful, so treacherous.
“Ned Stark a traitor?” Ser Jorah snorted. “Not bloody likely. The Long Summer will come again before that one would besmirch his precious honor.”
“What honor could he have?” Dany said. “He was a traitor to his true king, as were these Lannisters.”
[“]The day Lord Stark lost his head, I was there, watching. Afterward I went into the Great Sept and thanked the seven gods that Joffrey had stripped me of my cloak.”
“Stark was a traitor who met a traitor’s end.”
“Your Grace,” said Selmy, “Eddard Stark played a part in your father’s fall, but he bore you no ill will. When the eunuch Varys told us that you were with child, Robert wanted you killed, but Lord Stark spoke against it. Rather than countenance the murder of children, he told Robert to find himself another Hand.”
“Have you forgotten Princess Rhaenys and Prince Aegon?”
“Never. That was Lannister work, Your Grace.”
“Lannister or Stark, what difference? Viserys used to call them the Usurper’s dogs. If a child is set upon by a pack of hounds, does it matter which one tears out his throat? All the dogs are just as guilty.[”]
Daenerys has based her knowledge on Lord Stark, as she has on most Westerosi matters, on the stories Viserys passed down to her, stories Viserys himself received in probably simplified format when he was all of seven or eight. There would be no reason for Viserys to know about Lord Stark’s bastard son (especially when Ned himself did not even meet Jon until after the Sack of King’s Landing, which itself came after Viserys and the pregnant Rhaella were sent to Dragonstone), and consequently no way for Daenerys to have heard about him either.
Now, Jon might know, on an intellectual level, that Daenerys exists, or did exist at one time. He certainly knew his father fought in Robert’s Rebellion, and while questions about the war might have been discouraged by Ned’s sadness over his losses in it, Jon still recalls that his father had spoken “often” of “the peerless Robert Baratheon, demon of the Trident, the fiercest warrior of the realm”; Jon himself also knew that men “whispered Kingslayer behind [Jaime’s] back” without much confusion about the epithet on his part, which might seem to suggest that he had a basic working knowledge of the Sack of King’s Landing. Certainly, it was not a secret that the youngest of Aerys II’s children had escaped to Essos: the denizens of the Lazy Eel discuss the escape of the Beggar King and his sister (with only Davos getting her name right), while the acolytes at the Citadel do the same in the prologue of AFFC. Still, I would doubt Jon knows anything more than that Daenerys (if he even knows her name) was Aerys II’s daughter, and that she escaped with her brother to Essos at the end of Robert’s Rebellion.
All of this is to say, however, that there’s good reason to think they’ll learn more about each other before they actually meet. Perhaps not Jon about Daenerys in TWOW (he’ll be pretty busy with his resurrection, the coming threat of the Others, and of course his own parentage questions being finally answered), but I think very possibly Daenerys about Jon. She will, after all, have Tyrion as one of her primary advisers, who just so happens to have met Jon. As she comes onto Westeros proper, I think Jon will be quite interested to hear about dragons - what could be better to stop demons of ice than fire made flesh? - and the woman who birthed them, while Daenerys might start thinking a little more sympathetically about the Starks (given her vision of the Red Wedding), and perhaps that sweet blue flower in the Wall of ice …