Is there any proof that Quigley is lying?

[Optimism is my Phil-osophy]

VFD is a secret organization. Hector, who is definitely a VFD Agent, did not recruit the Baudelaires. In fact, Hector had a deck of lies and half truths ready to tell them. If Hector owned this deck of lies, other agents like all 3 Q sibligns could have a deck of lies. In fact, Olivia Taliban was one of the first not to use a deck of lies for the Baudelaire Sibligns, because she was their first scam. She believed they were VFD agents ... Besides, she seems to be a stray agent. She is a double agent, not because someone asked her to be a double agent, but simply because she decided to be a double agent, because of personal ideologies, and for being too long in disguise. This kind of thing stirs the mind of undercover agents even in the real world.
If you stop to think, all the skills the three Q singlins possessed seem to be VFD-trained agent skills. And do you really believe that between TAA and TEE Ducan and Isadora learned about VFD to the point of giving important tips to the three B? I think it is more likely that they already fully understood VFD, and decided to give some little secret information to the three B. Just as Hector did in revealing his self-sustaining balloon house.

Note: I am not saying that I really agree with this theory. I haven't stopped to think about her much yet. I'm just saying the theory seems plausible to me.

[Foxy]

I don't think Quigley is hiding anything. I think the events in the books sometimes fit together like imperfect puzzle pieces because the author is human and probably didn't expect his work to get picked apart to the point we have. He probably didn't know who the survivor was going to be in THH, and then decided when he was writing TSS.