This bugger of a beatdown all depends on which versions of these games we're talking about. Doom, of course, is up to its third incarnation, while Quake is on its fourth. The purist in me, however, wants to compare the originals of each title, and how the nameless main characters would fare against each other.
The Doom Guy is armed with mostly conventional weaponry: a shotgun, a chaingun, a rocket launcher, a chainsaw. He also, eventually, carries a couple of more exotic weapons: the plasma gun, and the BFG9000.
The original Quake Guy, meanwhile, carries similar weaponry (a shotgun, a super shotgun, a grenade launcher, a rocket launcher) plus his signature armaments: the nailgun, the super nailgun, an axe, and a lightning gun.
Now, armed with all of their weapons, the Doom Guy would probably just fire the BFG and blow the Quake Guy to atoms. Stripped of that possibility, however, the Doom Guy becomes just about the equal of the Quake Guy.
This smackdown would basically be a running, jumping, strafing affair with thousands of rounds of ballistics fired off before one of the two went down. It also would depend greatly on the available armor, health, quad-damage, and other goodies might be scattered about the arena. For example, if someone nabbed quad-damage and got close enough for a decent shotgun blast, the fight's pretty much over in a mess of bloody pulp.
Conclusion: with the BFG9000 in play, the Doom Guy wins hands down. Without it, the fight goes to whoever grabs a lucky shot. It's a draw.