I'd argue that Darkseid could beat the Q. During the Crisis on Infinite Earths, there was a being called the Anti-Monitor, (the Monitor of the negative energy universe) that launched an all out assault on the multiverse. The DC universe was composed of an infinite number of universes, and the Anti-Monitor blew up all but 5 of them. Yes, he subtracted from infinity and got a whole number.
Anyway, God, (like, the literal Christian god) got annoyed at the Anti-Monitor fucking up his multiverse, so he sent the Spectre, (God's wrath and vengeance given form) and the Spectre, fully empowered by God, punched the Anti-Monitor in the face and the shockwave from this was so powerful it shattered dimensions and was felt throughout the remaining cosmos including heaven and the Fourth World. The Spectre was put into a death-like state from the backlash. The Anti-Monitor was knocked out for a couple of minutes, afterwards, he was totally fine.
Later in the story, when the Anti-Monitor was about to win and wipe out all of reality, Darkseid, having come to the realization that the destruction of reality would kill him too, decides to intervene. All the way from his throne on Apokolips, in a different dimension, he fired his Omega Beams and almost completely destroyed the Anti-Monitor's body, weakening him enough that Superman, (the one from Earth 2, the really weak one from 1939-1956) was able to finish him off in one punch.
Darkseid is powerful enough to effortlessly defeat an entity capable of wiping out reality itself, an entity that God himself couldn't destroy even by fully empowering his avatar. Yes, that makes close to zero sense at like 10 points, but hey, that's DC for you.
Only thing i can say is Q is Q and there are many of them.