While I didn't realize Quirrel might be lying about his willingness to cooperate, the Memory books aren't in the restricted section and we haven't heard anything about him sabotaging Harry's attempts at taking over the universe since chapter 95 (except for some sort of deception in 100/101... "what a fiasco" is not something you would hear Quirrel say to himself audibly and honestly). Maybe he was in Zombie mode until 99? We wouldn't know, since we weren't in Harry's POV since 2 days after that chapter, although in his shoes I would have attempted contacting Quirrel as soon as possible.