Narratively, Super Quarterstaff Guy fights with, you may have guessed, a quarterstaff.
Can he do this with the Agility power (in the Athletic category)? Or does he have to have the Signature Weaponry power?
You would use Signature Weapon from what you’re describing. Now you could use different qualities for attacking using it.
That said, there's flexibility. If his schtick is "this quarterstaff is my ancestral weapon, I have practiced with it until it is like an extension of my own body, but give me a spear and I'm completely useless" then by all means go with Signature Weapon. If he's a martial arts practictioner who favors the quarterstaff but could use other things in a pinch, another Power might be more appropriate.
Or, run back a mix of the two. If he's especially practiced, or almost always seen with that quarterstaff, then Signature Weapon is an easy pick.
If Super Quarterstaff Guy often incorporates his Agility into how he is a hero, or in how he fights in general, then you could also easy pick Agility.
It's really a case of how you see them being a hero, even beyond situations where they are beating someone unconscious with the quarterstaff.
Also look at the RED zone abilities for the 2 diffrent powers. If one has something you want and the other has nothing for you you might only need one.
Big difference is the narative logic of how you use Agility or Signature Weapon for non attack basic actions. a guy with a big stick might boost/hinder/overcome differently then someone with balance and reflexes and some situations it might not really make sense to use one of them.