Q usually prefers to avoid meeting alternates of himself. It's very frustrating for an entity that prides himself on his strength of will and his ability to forge his own path to be reminded how different he could have been under other circumstances. When the nutcase from a parallel universe shows up ranting about Picard, however, and declaring that Q's own universe is obviously doomed, the mixed feelings of pity and disgust drive him to try to help.
The nutcase believes that every alternate version of Q has to be involved in a relationship with that universe's version of Picard. (Q doesn't manage to avoid making the sarcastic comment about what about universes where there is no Picard. The nutcase claims there is always one, although sometimes he's not bald, French, human, named Picard, or male. Q does manage to avoid asking how, then, he could possibly be defined as Picard.) He believes that if he doesn't manage to form a relationship with his version of Picard, his universe will be destroyed, and that Q's universe is obviously doomed, because Q is destined to have a romantic relationship with Picard and it will cause the timeline to be resorbed into mainline if he doesn't.
This is not the most insane thing Q has ever heard, but it's the most insane thing he's ever heard from a fellow Q, and certainly he's horrified to hear it coming from his own alternate. Quite aside from the absurdity of a romantic relationship with Picard, the notion that there is any such thing as destiny, that something as minor as a romantic relationship could possibly define whether or not a timeline is "real" enough to avoid resorption, or that a timeline could in fact be resorbed into mainline for being too different from other timelines, are all ideas that are simply wrong, and a Q should know better. Clearly the strain of being in love with an unworthy mortal who couldn't even be bothered to reciprocate despite the great honor of being loved by a Q had unhinged the poor entity.
So despite his natural instinct to avoid asking the Continuum's aid at all costs, Q calls them in to help him capture and rehabilitate his alternate. An insane Q really can't be allowed to roam spacetime freely. Of course, the alternate resists-- he may be insane, but he's still an alternate of Q, the most stubborn entity in the Continuum, and he won't allow his perception of reality to be forcibly overwritten even if it is plainly psychotic. So they lock him up in a comet to keep him from doing any damage, and ignore his hysterical ranting about how both their universes will be destroyed.
A thousand years later Q lets his alternate out. Congenitally unable to resist saying "I told you so," he indulges himself by pointing out in excruciating detail that Picard is over nine hundred years dead and neither of their universes have been destroyed. The alternate ignores him and promptly returns to his own universe, jumping back through the intervening thousand years to intersect his home universe at the time he left it. Q sighs, realizing the nutcase still believes he has to get involved with Picard to save his universe, and decides to write the whole thing off and pretend it didn't happen.
Author's Note: Any resemblance between this AU and the plot of a well-known Picard/Q fanfic is... entirely on purpose, I assure you. I mock because I love.