I just rushed through St. Paulia's character quests without really paying attention to medals or bothering to kill all pests. Now that you mention it, I'm very curious as well. If you ask me, I'd say no. I doubt that you can get a second reward by killing all pests/bosses, since you can't replay the character quests. That would just be cruel. But feel free to correct me if I'm wrong, anyone.

Then there's the regular reward for clearing the character quests, in St. Paulia's case I think it's a necklace? But that's awarded regardless if you kill all pests or not.

In none of the current Character Quests, it makes a difference if you get any amount of medals. You only get a character specific Earring for the character when you finished her final and third quest.