The fact that Patrick Stewart played both Picard and Charles Xavier has inspired me to do several X-Men crossover fics taking place in the X-Men movieverse or some AU variant thereof. Also being generally obsessed with X-Men, there's at least one, and I may do others, where the crossover is more X-Men comics inspired.
Similar Features v. 1.0: I am absolutely sure I will expand this at some point, and when I do it will most likely be a slash fic, though this initial take on the idea is relentlessly gen. This is an X-Men Movieverse/Star Trek crossover, taking place after X2 but no real spoilers. Charles Xavier has in his basement an entity who knows Abbott and Costello, thinks he's a 24th-century Frenchman, and is afraid he might accidentally eat the sun. Life is never boring chez Xavier's.
Where No Mutant Has Gone Before: Another X-Men Movieverse/TNG crossover (I swear they're just too damn easy). This is a remix of Andraste's "To Boldly Go", a drabble that features Charles Xavier having switched places with Captain Picard. In this remix I expand the concept to explain how this happened. Three guesses as to what deus ex machina I use. The first two don't count. :-) Sooner or later I might write Picard's POV of these events. :-)
Prometheus In Chains (incomplete): This story is a "Q loses his powers" that doesn't depend on either "Deja Q" or Q doing something stupid, and in fact it can be interpreted best as a spinoff of "Civil Disobedience". It's also going to be quite NC-17 with m/m and f/f, eventually, but right now it falls under the "dream sequences" clause. It also has X-Men characters pretending to belong to the Star Trek universe, filling out the cast of characters on a colony world.
Tetrahedron (incomplete): I haven't posted this anywhere but here yet, so if you actually surf to this page and find this, bonus for you! X-Men First Class fusion with post-TNG era where all the First Class characters have been re-mapped to be characters within the ST universe (so Raven Darkholme is a Chameloid, Ema Fros is an albino Betazoid, etc.) Charles Xavier, the young telepathic captain of the Excalibur, has been struggling to free himself of the shadow of Jean-Luc Picard, the man he was cloned from. But when Q taunts him regarding the possible rebirth of the Borg, he has no choice but to talk to his progenitor, despite both of their reluctance to deal with each other. Will be eventual Xavier/Magneto, contains heavily implied Picard/Q.