Quigley Quagmire at the beginning of The Austere Academy was thought to have died in the fire that killed his parents until Violet and Klaus Baudelaire met him in a cave with the Snow Scouts in The Slippery Slope. He and Violet Baudelaire clearly have some romantic interest in each other; at one point, Snicket implies that Violet and Quigley kiss (as they climb a waterfall, Violet comments on the view, which Quigley agrees looks beautiful; however, he is looking at her when he says this, giving the reader the impression that he is saying that she is beautiful instead of the view). Snicket then decides to break off at this point and let Violet have some privacy. However, he tells the reader that, throughout the rest of their climb, they had "small, secret smiles," and, when Quigley expresses surprise that they've been climbing the whole afternoon, Violet reminds him (while giving him a shy smile) that they have not been climbing the whole afternoon. Also, when the siblings are separated from him, Violet is the only one to start weeping. As well as this, throughout the next book, Quigley is always described as "a cartographer—someone who is very good with maps, and of whom Violet Baudelaire was particularly fond."
His mother hid him in a trapdoor during the fire to protect him. He was under there for a few hours, then he found that it was a tunnel. It took him to Dr. Montgomery Montgomery's house, after the Baudelaires had been there. The house was empty, proving that the Baudelaires had already left and Dr. Montgomery was already dead. This also implies that the Quagmire house burned down after the Baudelaire mansion. Then, Jacques Snicket arrived, but left for Paltryville.
At the end of the book, the Baudelaires and Quigley escape Olaf, but the icy Stricken Stream melts, and Quigley gets washed away by the current of the river into a separate tributary. He apparently survives, as he sends the Baudelaires a telegram with a hidden message in it in The Grim Grotto, telling them to go to Briny Beach. He also hides the word 'violet' in the message, which seems to have nothing to do with telling the Baudelaires where to go -- Violet suggests that he just wanted to write her name, a further implication of their romantic involvement. Violet then runs to the taxi cab, saying Quigley's name under her breath (it is earlier stated that she felt as if she had been whispering his name to herself for days), before shouting it. He is on the "good" side of the schism.
Like his siblings, Quigley has an interest related to writing, in his case cartography.
In The Penultimate Peril, Kit Snicket tells the Baudelaires that Quigley and Kit were 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 the series, Kit informs the Baudelaires that Quigley did indeed manage to meet up with his siblings, shortly before the eagles popped the balloons holding them up. Everyone inside the mobile home was sent toppling downwards, destroying the Queequeg directly below them. Duncan, Isadora, Kit Snicket, Captain Widdershins, Fiona, Phil, Fernald, Hector, Ink and Quigley were all left stranded in the water, before the large question-mark shaped object, The Great Unknown, appeared below them. Kit and Ink managed to escape, while everyone else wanted to take a chance with the unknown and were pulled under, to their rescue. It is unknown if they are alive or not, but in The End it mentions 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.