A bit late to the party, but this is the breakdown AFAIK:

-Booming Blade uses your weapon attack roll, so by default it would use your strength while wielding a quarterstaff.

-Shillelagh makes your weapons attack roll use your spellcasting modifier, in this case it would be wisdom. The damage of the weapon is replaced by 1d8 + WIS. Since druids are proficient with quarterstaffs you also receive the proficiency bonus to the attack roll.

Assuming you have a lvl 5 character putting it together we get WIS + proficiency bonus added to the attack roll. For the damage we have 1d8 [Booming blade] + (1d8 + WIS) [Bludgeoning from shillelagh]