Not quite yet they don't. They played against Albion Rovers today in the national stadium and will continue to play there for the rest of the year before moving to Lesser Hampden.
Queens Park used to play in what is now known as Cathkin Park before Third Lanark used it as their home stadium.
Cathkin park was the 'second Hampden' before they built what is now the national stadium (the 'Third Hampden'.