A 24 (cont) From PHB p. 78.
"monk weapons, which are shortswords and any simple melee weapons that don’t have the two-handed or heavy property."
Being versatile doesn't disqualify it as a monk weapon. A qtr staff is a simple weapon that doesn't have the two-handed or heavy property. There is no restriction that says it must be wielded one-handed that I know of. Thus by RAW monks can (and generally I'd say should) wield a qtr staff with two hands, particularly since their unarmed strikes don't require a hand free. (Unless and until someone can show us such a restriction.)