Its easy, once you get your Qatari residence permit, you may drive the car yourself if you still have UAE visa, and if you cancel you visa, you may have to cancel you number plate in uae for your car, and then get an export numberplate for Qatar, and put that number plate on car and drive to Qatar, its 6 to 7 hours drive, but you will have get Saudi Arab visa for transpassing before travel.

Once you car is here, you may goto to traffic department and apply for car registration, there are some charges for car registration.