From MQG's unique:
"... Whenever any other Jedi ally attacks during Master Qui-Gon's turn, he assists (once per turn). ..."
However, until his first turn if any Jedi ally attacks out-of-turn (including counters) he will assist once on that turn. The clause about only assisting on his turn doesn't take effect until MQJ takes his first turn.
This is most easily observed running him under Jedi Master Luke Skywalker or Jedi Knight Revan (as long as he goes not first on that team) and using the special granted abilities from their leaderships on a different Jedi who isn't MQJ. You will see MQJ assist even though it isn't his turn.
I posted a different bug about a month ago with Reva involving MQG where her GAC leadership could apply deathmark on enemy chars outside of their turn, but given the above I am now wondering if the source of that bug is actually MQG himself. Until his first turn he seems to treat every turn as his own turn, so perhaps the root cause is the same.