Yeah, I don't see this happening. Why would Q *ever* subject himself to serve as a lowly duty officer on anyone's ship? Any Q should be far too haughty and egotistical to even consider the idea.
Yes, I know, its IP, in that the John DeLancie Q did actually do this, but he was stripped of his Q-powers and rendered mortal. He was thus no more powerful than anyone else on the ship (and arguably less useful, as he was completely untrained in the use of any equipment), and so there wouldn't really be any special reason to seek one out as a crewman.