You can obtain a "Temporary Qatari Driving License" valid until validity of your Business or Visit visa.

Just go to Traffic Deptt (Madinat Khalifa) with eye test report (to be done at any authorised eye testing shop), a letter of your Employer, Passport Copy and your original UAE License. It would cost you around 200 QR.

You can obtain Permanent Qatari Driving License once your Residency Permit is done. You need not to go personally and can send anybody to get your Temporary License converted to permanent (valid for 5 years). It would cost you again around 200 QR.

Dont forget, Traffic Deptt do not accept cash payment and hence you would have to pay by Card.