Nothing is known of what happened between them, but the narrator thinks that he can make a few speculations based on their magnanimous and almost super-human nature.
He thinks that Vere did not flinch from telling Billy of his own role in the case. He thinks that Billy could understand Vere's dilemma, and that he knew that in sentencing him to death, Vere acknowledged that Billy was not afraid to die.
He thinks that Vere's composure may have broke, that he may have come to look on Billy as something of a son.
But in the end this is all speculation. There is no way to actually know what happens.
The senior lieutenant is the first to see Vere when he emerges from his conversation with Billy.
He can tell by looking at his face that the discussion was harder on the condemner than on the condemned.