You won't be able to attack with both weapons in the same round; in order for you to get a bonus attack without the class feature or feat, both weapons must be light. That's demonstrated in the section on two weapon fighting:
When you take the Attack action and attack with a light
melee weapon that you’re holding in one hand, you can
use a bonus action to attack with a different light melee
weapon that you’re holding in the other hand.
Since quarterstaves are versatile, however, they can be wielded with only one hand. Just as you could attack with a quarterstaff while wearing a shield, you can attack with a quarterstaff while bearing another staff.
From the DMG:
Unless a staff's description says otherwise, a staff can be used as a quarterstaff.
From the Staff of Power description:
This staff can be wielded as a magic quarterstaff that grants a +2 bonus to attack and damage rolls made with it.
From the Staff of Striking description:
This staff can be wielded as a magic quarterstaff that grants a +3 bonus to attack and damage rolls made with it.
Since you aren't intending to attack with more than one staff in a turn, and the staves in question can be treated as quarterstaves, you can indeed have one in each hand while you fight, using one for melee attacks and the other for its spells.