Maybe I'm being pedantic, but a quarterstaff is a staff .. as in part of the weapon group staff.
A wizard wielding a magic orb, staff or wand can add its enhancement bonus to the attack rolls and damage rolls of wizard powers ...
Now, a magic quarterstaff may not necessarily have been enchanted to be an implement, but it's a staff, it's magical, and it has an enhancement bonus.
So I think the implication is you can, seeing as as any implement staff is 'just' a magic quarterstaff for anyone not trained in its use as an implement.
Going by the implement staves section, they can be fashioned either as a quarterstaff or a walking staff (also imbued with arcane enchantments, but meh) ..
now if a nonmagical implement costs the same as a quarterstaff, and can be used in melee exactly like a quarterstaff, I would have to assume it is a quarterstaff.
If a magical staff that was cut as a quarterstaff before being enchanted is good enough to be a staff of defense, then why wouldn't any old sturdy-enough-stick do?