Dear Roya, i don't think you can have both since in your working visa in Qatar you can enter UAE anytime you want (GCC rules). Besides, if they don't allow that you hold 2 driving license in the same time (it happened to me personally), I had to give them my UAE driving license in order to transfer to Qatari driving license, definitely they will not allow 2 residency visas.
best of luck... if you need more info, you can send me message or email. regards