No, Qatari's get extra salary from the government, not from QR. All Qatari's get a share of the wealth of the country.
As for leaving for the Air Force it's because they get an even higher salary and their job consists of them flying around in a mirage or alpha jet for a maximum of 2 hours a month and then going home and relaxing with their gigantic paycheck. Definitely beats minimum rest turn arounds and the crap that goes with commercial aviation.