As it stands, only each classes first companion actually has a questable story. The others all have story points and talking background, but no quests (or very few. I think Torian gives Bounty Hunters ONE quest).
This is actually off. For the JK, T7 doesn't have any quests. Meanwhile the JK's second companion has two. I know off the top of my head that the JK's fifth companion(and I'm sure other companions as well) has a quest in the datamined files, but that quest is not in the game currently.