Sorry, I don't mean Major Boothroyd and Richard Hatch or whoever his reimagined version is.
Maybe I should have specified Star Trek
series or film at the top.
I think Nerys Myk
took it in that direction because your initial question is so obvious that you answered it yourself in your own OP by trying to allow Apollo a handicap. Q would win, with or without a handicap, unless the handicap were someone who could best Q themselves - another Q, "God" from ST:V (as depicted in the novels before he lost his head), etc. And in that case, that's less a handicap for Apollo and more just Apollo being a spectator while Q and whomever fight it out.
I'm now imagining that giant hand making a wanking gesture.