No private flying here. I'm 99% sure of that. Even Qatar Airways flights waste huge amounts of fuel skirting the coast and avoiding any direct routes that would take them over the US base. Qataris who want to learn to fly but who can't get into a military or QA program take themselves off to the States. (Even yet!)
If you want to double-check, try Doha International Airport and Qatar Aronautical College, both easily Googable, and make further enquiries.