Dr.Manhattan is an atomically powered being who can pretty much do anything he wants. His weaknesses are that he is no longer human and therefore is loosing his emotions and has become too logical. I think that his precognitive and postcognitive abilities are limited to his lifetime, therefore he can't exactly travel past his birth.
Q is an omnipotent being that can pretty much do whatever he wants. His weaknesses are arrogance and his actions are sometimes too impish that they can be ignored easily.
Q's usual antics won't work on Dr. Manhattan since he has no emotions and is too curious to be affected in the way that Q expected. I am assuming that they are both on the same power level, which is not too far fetched since Dr.Manhattan can travel to other galaxies thus leading me to believe that he can do whatever Q can. That being said, I had to look at the weaknesses of each combatant. Q's seem to be vulnerable to imprisonment by other Q, which I guess means that something about that asteriod in that Voyageur episode could contain him. The problem is that Dr.Manhattan wouldn't know this and therefore I can't consider that. Dr.Manhattan's weakness is that he isn't creative, he is too logical and Q is very creative and I think he could trap the doctor in the same manner as the rebel Q. Q is clever enough to trick Dr.Manhattan just like Ozymandias did in the Watchmen.
winner: Q
Still undefeated