One is a brooding emo with the vocabulary of a 90's grunge singer...

The other has serious mental issues...

Together, they fight crime the forces of EEEEEEEEVIIIIIILLLLL!!!!

...sorry, couldn't help myself. But that is pretty close to what it would happen.

First and foremost, Sephy is a loner; what does he gain from joining with someone he, up until later where he gets his fetish he doesn't treat as anybody? Or someone who constantly broods over being alone and misunderstood (though he would probably ka-stab Squall just because of that)? Or worse, someone with the idea he's a specific kind of hero?

Squall and Seifer wouldn't specifically join; they are rivals, and by the looks of it, they can't stand each other (one is the anti-ego of the other, and they tend to repel each other). He wouldn't have a reason to join or fight Cloud, either. Perhaps if Sephiroth revealed himself as a sorcerer or something, but wielding a big sword doesn't usually translate to "incredible skill in magic".

As for Cloud...well, he's kind of a loner himself, and also thinks he's a bit misunderstood (following a mercenary profile at the beginning), so he wouldn't join either side.

So, in the end, they would somehow notice something is wrong; specifically, since there is no unifying threat that would force four equally disparate beings to work together, or even oppose themselves aside from the established pairs, but that could be resolved separately and with no need to fight each other.

That, or play Dissidia and get another of the swordfighters aside from Sephy as your "Seifer". That should give you an idea of what it is to have all four together. Aside from that, not much. Not even "kill 'em all, Sephy" since that would be a waste of time and not a reaction.