TYere planning to meet up with the three children, but he received word from his other two siblings that they were being attacked in the sky. Kit says that he stole a helicopter to help them and Hector, and would do this by constructing a huge net.
At the end of wn and were pulled under, to their rescue. It is unknown if they are alive or not, but in The End it is mentioned that the Quagmires are "in circumstances just as dark [...] as the Baudelaires'." Kit also mentions that the last she heard of them was one of the Quagmires calling out Violet's name. Sunny asks which sibling called it, but Kit didn't know. It is most likely that the sibling who called out Violet's name was Quigley.
Quigley's first name may be a reference to Lillian Fox Quigley, author of The Blind Men and the Elephant, a children's book based on a poem of the same name discussed in The Penultimate Peril.
Quigly is then not seen again, after The Slippery Slope, as it's know for him to have drowned in the ocean. Later it says in Penultimate Peril he is looking for his siblings.