Grace in a Vacuum by Marcelo. Stargate Atlantis with Q. Rodney McKay goes through a gate and ends up in a completely deserted Atlantis, with bizarre hallucinations as his only companions... except, of course, they're not hallucinations at all. Rated PG. Recced.
In A Galaxy Where No Man Has Gone Before by Merfilly. Star Wars/TNG, explicit slash. Q isn't exactly in this -- it's a Picard/Qui-Gon Jinn fic -- but the story is more or less a sequel (though it was written first) to Merfilly's "Misunderstandings", recced below, and Picard's backstory with Q is one of the drivers of the plot. Contains explicit consensual m/m sex and implied rape. Rated NC-17.
Information Indiscretions by Anya. TNG/Hitchhiker's Guide. Q is only a minor character in this one. Data meets Marvin from Hitchhiker's and becomes infatuated. Strongly implied P/Q relationship but the entire story is PG-rated.
Colorblind: by Duck. TNG/Firefly. River follows a Crow to the TNG universe, where she learns that she is much more sane out of her home 'verse but much less powerful, and that Crow's other name is Q. PG. Recced.
I Shall Be Made Thy Music by Andraste. TNG/Battlestar Galactica (new). God shows up and offers Gaius Baltar a chance to undo his terrible mistake... at a possible price. Gaius isn't quite sure this is actually God, though. PG. Recced.
In The Shadows by Eren. TNG/Babylon 5. In a universe where a very, very dark version of Jean-Luc Picard is Emperor, courtesy of Q, Lyta Alexander is trying to find a way home... but unfortunately finds Picard first. Picard/Q, though not exactly "our" Picard. Dark, non-explicit torture. Rated R. Recced.
There Is Nothing Like The Truth of the Fool: by Leyenn. TNG/Babylon 5. Before the coming of the Shadows, Morden meets Q in a dream. PG-13. Recced.
Res-Q by Astrogirl. TNG/Farscape. Q finds John Crichton too entertaining to let him stay dead. Humor. PG. Recced.
Meddeling: by Nemesis. TNG/Doctor Who. Milan is burning and the Doctor runs into someone who's as out of place as he is. PG.