Many quilters who come to Doha wonder where to buy fabrics. You will find some shops in the Suq al Waqif, with parking spaces and restaurants nearby, and sometimes you can even find cotton in these shops. You never know!
Your best bet when shopping for quilting cottons are the Souk al Asseiri and the Souk al Dira'a (sometimes called The Marble Souk, for it's marble floors). Above the map is a photo of the Souk al Asseiri, some people call it the red and white building, some people call it the building with pillars. It is mis-labled on the map, so I put a blue marker where it really is. The Dira'a souk is where the red marker is.
The big problem in this area is parking, especially now with the huge area under construction. Many people park across Grand Hamad at the Suq al Waqif and cross Grand Hamad and walk to these souks. My secret is to go early in the morning - like around 8:30. There is almost always a spot, and you get in and out before it gets too hot. You can also ask for the "first customer" discount. ;-)
Batting (wadding) is always a problem. Many quilters stock up while home for the summer. Some buy polyester batting from upholstery shops. Jarir bookstore used to carry batting, but hasn't had it for a long time now. Classic Quilts in Dubai usually has batting, and now there is a budget airline from Qatar to Dubai.
Across the street from the Dira'a souk is one of the Al Rawnaq stores; this one has all kinds of notions, yarns, buttons, fake jewelry, etc, but most are not a very high quality.