하다 is a irregular verb. (Here 다 is just a basic ending to make the verb stem into the verb.) When 하 conjugates with the conjugative ending(-요, *-도*, -서), 여 is added between the two. 하 + 여 + 요 = 해요, 하 + 여 + 도 = 해도, 하 + 여 + 서 = 해서. It's called "여 불규칙(irregularity)". Fortunately, 하다 is the only verb which follows the rule. -도 = "if" or "even if". We actually say 질문 하나 해도 돼요? or 뭐 좀 하나 물어봐도 돼요?

This sentence means if I can ask you a question,but very politely. 해 is the changed style of 하다,it means do. 도 is used to ask for the opnion of the other one.