Is there a way to trigger new conversations with my squad mates without doing missions? I've run out of missions to do before the Reaper IFF and I'm almost positive I haven't seen all the dialogue for some of my squad mates.
No - dialogue is based on where you are in the story.
If you feel you're missing some dialogue then the only way to get back to this would be to replay through the game. It's also not possible to experience all of the dialogue in a single play through either - since every choice made in game has consequences (but, for example, if you play the game entirely paragon, you miss out on all of the renegade dialogue and vice versa).