If you fancy a game on sand using a plastic mat when you are on the fairway. QP have courses at Messaieed and Dukhan. Membership is reasonable priced but not as cheap as for QP staff.
The Doha Golf Club is a very good course but can get a bit pricey if you play regularly.
There is a pro shop at the DGC and there was one at the Messaieed GC.
Olympic Sports also sell clubs and other accessories I believe.