I watched episode 3 again last night. Davos told Dany that Jon had taken a knife to the heart for his people and would have added that he had died if Jon hadn't stopped him. Why doesn't Jon want Dany to know that? It's not a secret, at least among the Night's Watch.
I think Jon is worried that it would make him seem more like a threat to Dany, who doesn't trust him. But she definitely noticed, because she asked Tyrion about the knife in the heart comment later. At some point, Dany is going to find out, and it may play into the narrative that together, she and Jon are the prince that was promised to save the world, especially when she also sees the threat from beyond the wall for the first time.
I think Jon is worried that it would make him seem more like a threat to Dany, who doesn't trust him. But she definitely noticed, because she asked Tyrion about the knife in the heart comment later. At some point, Dany is going to find out, and it may play into the narrative that together, she and Jon are the prince that was promised to save the world, especially when she also sees the threat from beyond the wall for the first time.