If Q and Apollo got in a fight...
...who would win?
Since Q appears to be omnipotent, and Apollo was stuck on a rock for centuries, we'll let Apollo have one (1) additional partner on his team. Any character from any series or film is eligible (even another member of the Q Continuum).
I'll start by giving him Trelane. Since both seem to require some sort of amplifier, Q can make those tools disappear, and wins.
Dr. Howard, Dr. Fine, Dr. Howard: For Duty and Humanity!
--Men in Black, 1934.