First of all, I doubt you will be allowed to have Qatar Driving license with acquiring Resident Permit and will require letter from your sponsor, to appear for the test.
Sorry to shoot down your eagerness.

If, this is not the case you will have two options :

Option 1 ( Difficult)- You can approach Garaffa Police Station, with a letter from your sponsor and if you are lucky, by paying QAR 250, your driving test will be conducted, in your car, and chances of clearing it is remote.
Option 2 - Attain a letter from your sponsor, apply for driving training ( beleive me, you will require it to clear the test), pay approx QAR 2,500 for traing and getting immediate date is difficult. Clear the test and get the license.
Option 3 - Go to any other GCC country, Europe, USA, Singapore, Get a driving license from there and then you can get direct driving license.
Good Luck.
Sandeep