In what regards? Like their previous duels and stuff like that? It's possible, but I don't know if it's very likely, because Mihawk doesn't seem like much of a social person from what we know of. But maybe his lips loosen when he drinks. It all depends on what kind of relationship the two formed over the time skip. If Mihawk saw him strictly as a student, and one day future opponent, then probably not. He just doesn't seem the type to open up to people.