I don't know if it's a matter of "Who could beat up who" but more "Who's more of a badass". I mean, sure, Q is an all powerful semi-godlike entity, but on the other hand.....Darth Vadar? I mean. It's Darth Vadar.
Ed: Q is really the only one I find problematic really (if we're looking at it as a straight up fight). Adama vs the Doctor? The Doctor isn't invincible. Sure, he can travel through time. But he's vulnerable to a bullet in the face. And even Q, for all his power, usually has to play by a certain set of rules. Hence why, if it came down to Picard vs Q, Picard would probably win.
As for the Matrix guys....I dunno. Probably in the Matrix since Agent Smith is included. This is why I'm thinking of it more as a "Who's more awesome" kind of bracket.
Anyways, some surprising first picks of mine (I am surprised at my own choices, or they were really hard to make):
Sarah Conner > River Tam
Kaylee > Scotty
Ripley > Hiro Protagonist
Borg > Daleks
Darth Vadar > Q (but really Q let him win to teach him a greater lesson)
Paul Atreides > Ender Wiggin
and then some really easy ones
T-100 > Darth Maul
Wintermute>HAL (sorry Halforums)
Spock>Neo
Pickard>Kirk
Han Solo >>>>>>>>>>>>>>>>>>>>Morpheus
And then there's the ....I have no idea.... category
Deckard ?? Mal
The Gallifrey Region is going to be the most difficult by far, by I'm putting Han Solo as the final winner......or maybe Picard????
Magrathea will either go to Deckard or Mal
Caprica will go Ripley or Hiro
Empire will probably go Darth Vadar, but Q or one of the princelings (Wiggins/Atreides) could squeak from behind. Really if Q doesn't go out early he will dominate all the way to the end, unless there is a Picard v Q round....I doubt Q would take Picard.