So I acquired a set of Q faction cards and I think Admiral Q has a lot of potential. He is -1 to skill, but there are ways to mitigate that. There are lots of skill boost cards out there, Picard with Riker from the Pegasus and Admiral Q would still have you at skill 11.
I feel that Romulans could benefit a lot from Q. Being able to basically come about with a D'Deridex is nice, it hels mitigate the lack of a rear arc on the Warbird. Salatrel seems like a good fit for Q because with her captain text you would still shoot at 10 with an enemy in your arc. The biggest down side for Romulans is that to take Admiral Q you have to give up Admiral Mendak

which isn't always worth it.
Now if you don't have a small stack of Dorsal Phaser Arrays laying around Q would be a help to the Federation too. Thinking along the same line as above the Doctor is a good captain to go along with Admiral Q. Again he doesn't care about the drop in skill because he is probably your lowest captain anyway and again the Doctor's text takes him back to 10.
I like this
22 USS Enterprise
01 The Doctor
01 Admiral Q
02 Type 8 phasers
02 Systems Upgrade
27
Not terribly expensive. You don't care about Aux tokens so the doctor can still use his text. If you get a stack of Aux tokens from using Q the ship can still take an action. Would be to pull a 2 turn, then 180 with Q to hit someone, like those fancy turns in X-Wing