Like many other buildings in the story, the Quagmire Mansion has a secret passageway, this one leading to Montgomery Montgomery's House. Because of this, the mansion must be located at or near Lousy Lane, and is probably not in the City.
This passageway was used by Quigley to flee the mansion at the time of the Quagmire fire. Both of the Quagmire parents perished along with the burning mansion.
The mansion was burned down supposedly by Count Olaf, who wanted to gain its fortune, the Quagmire Sapphires. He doesn't manage to do so, because we see that Olaf has Duncan and Isadora "in his clutches".
In the TV series, a mysterious woman is seen burning it down with a laser device that can set fire from a distance.