[What has Chekov-that-is-from-a-later-canon-point-than-this-one-as-of-this-thread been telling you, Kirk? (Probably plenty. Given a little more time in space together to get past the captain-crew relationship barrier, Chekov would (will) absolutely confide in Kirk.)
Mostly he doesn't know what to tell Kirk about Kitty because that's a confusing thing that involves a long story that isn't appropriate when there are hungry raptors around. This is stalling.]