by Kamakaze
The above answers are correct regarding the timing but I will add that even if it was possible to to make a character quest twice, it would have NO effect on your total willpower for the quest.At the end of the quest phase when you total your willpower, you simply add up all your willpower on the characters which are "questing". Assigning over and over to the quest would achieve nothing. (except trigger "when a character quests" triggers)
This is similar to attacking, where all the attacking characters total up their attack values.