The famous Quagmire Sapphires are a collection of valuable gems, likely sapphires, passed down through the Quagmire Family. In the Netflix series, it is said by Count Olaf that him and Esmé Squalor will have 1,000 Sapphires.
They also attracted the attention of Count Olaf. He strongly desired the Quagmire Sapphires and as is common of many of his other endeavors. Olaf most likely went about getting them by murdering Mr. and Mrs. Quagmire. However, it is unknown whether or not he was responsible for the Quagmire fire. In the Netflix adaptation, a woman uses a laser device to burn down the mansion. This could be Count Olaf in his disguise as Shirley by the pink clothing.
At the end of The Austere Academy, Olaf set his sights on using the Quagmire triplets to gain access to the sapphires. He kidnapped Duncan and Isadora and hid them in a cage at the bottom of an elevator shaft, then later in a statue of a red herring, in The Ersatz Elevator. They were then hidden in the Village of Fowl Devotees' Fowl Fountain throughout the course of The Vile Village. With the help of the Baudelaire children, the Quagmires were eventually able to escape. Afterward, Olaf's interest in the sapphires seemed to wane, and the current status of the gems is unknown.