My dear,
I did the same from 1 month but with my valid UAE licence, follow the following steps:

1- NOC letter from your current sponser here in Qatar.

2- Take your Bahrain licence with copy of it.

3- Eye test. . you can do it in traffic dept. for free but many people will be ther and you have to wait, or do it at any optics shop for 30QR.

4- Your ID here in Qatar + copy.

5- 2 pesonal photos with non white background

6- submitt all these documents in traffic dept. and they will give you Qatari licence without taking your Bahrain licence.

You will pay 250QR. there by credit card or ATM, CASH is not allowed

good Luck