I'm not sure I quite follow.
The icon has appeared, with no new content, immediately after 8-9 different major story developments, and I've clicked Talk immediately after it appears every time, and each time there has been no new dialogue available in the tree.
I think I've only ever actually been "on" my ship and able to talk to my companions there like three times in my playthrough, but there has never been any new dialogue available on the ship in particular.
There also seems to be fairly little "reactive" content from my companions. They never actually start any conversations about what's going on except as a direct result of doing that companion's personal quest line.
I wouldn't be surprised if I'm missing out, but I don't know how to be sure.
Edit: Okay, upon completing the main story, I checked out youtube for relationship content with the characters, and I've missed out on more or less all of it as far as I can tell. That's honestly extremely disappointing.