No, it has nothing to do with gifts. The icon comes and goes even if you stand there doing nothing. It's just borked. Like I said, just right click on them and they'll tell you if they want to talk or not.
By the way, a good portion of the conversations are gated by story progression, so even though in theory you should open up a new convo every 1,000 pts or so of affection, they won't actually trigger until you reach the appropriate point in your class quest. So you could grind a companion up to 10,000 affection at level 1, but you wouldn't get all the convos until their pre-allotted times (although you would get several back-to-back at various points).