What happens to the Quagmire?

Answer 1

get swallowed by the great unknown

Answer 2

During the Austere Academy (5th book) the Quagmires get stolen away by Count Olaf. In the Ersatz Elevator (6th book) the twins get stolen away again from Olaf after he hides them in the Statue and takes the item during the auction. In the Vile Village (7th book) the Quagmires are found by the Baudelaire's in the fountain in the Village of Fowl Devotees. Just before the Baudelaire's are safe, Esme shoots the ladder of the rope to Hectors flying home which Isadore and Duncan Quagmire are on, and the twins and Baudelaire's are separated again. During the 10th book the Quagmire's are noted to be seen in the air, and may have been attacked by the eagles that belonged to the woman with hair but no beard.