I've been wondering the same thing. You also need to look at the Lajatang and Three-Section Staff when considering this question (all two-handed special monk weapons).
Normally a quarterstaff is treated as a double weapon, and therefore you'd use the rules in the PHB:
I'd read this to say he gets normal strength bonus to damage on all attacks while using FoB. That is, it is like using two special monk weapons. However, while not using FoB I'd say he gets 1.5x Str bonus to damage.
However, with a lajatang and 3SS, I'd say he gets 1.5x Str bonus to damage on all attacks (whether using FoB or not) since they are not double weapons.
I think that I recall Caliban ruling that a monk would only get normal Str damage using FoB with a quarterstaff.
I'm not convinced yet that they shouldn't get 1.5x as per the lajatang and 3SS, but I could be swayed either way.
Normally a quarterstaff is treated as a double weapon, and therefore you'd use the rules in the PHB:
A monk may also use the flurry of blows if armed with a special monk weapon (kama, nunchaku, or siangham). If armed with one such weapon, the monk makes the extra attack either with that weapon or unarmed. If armed with two such weapons, she uses one for the regular attack (or attacks) and the other for the extra attack. In any case, her damage bonus on the attack with her off hand is not reduced.
I'd read this to say he gets normal strength bonus to damage on all attacks while using FoB. That is, it is like using two special monk weapons. However, while not using FoB I'd say he gets 1.5x Str bonus to damage.
However, with a lajatang and 3SS, I'd say he gets 1.5x Str bonus to damage on all attacks (whether using FoB or not) since they are not double weapons.
I think that I recall Caliban ruling that a monk would only get normal Str damage using FoB with a quarterstaff.
I'm not convinced yet that they shouldn't get 1.5x as per the lajatang and 3SS, but I could be swayed either way.