I think they only grant CPL conversions if you already have a job offer in the UAE.
It's a more or less standard license validation process. You'll need a Letter of License Validation from the FAA to allow for the GCAA to request validation of license, ratings and medical. You may need an FAA Accident/Incident Report too. They'll also want to see signed logbooks or certificates of your flight experience. Once that's done you'll need to do a full initial CLASS 1 medical once in the UAE and an Air Law written exam. If you already have a job offer in Dubai the airline can accelerate the process I'm sure but you will need to obtain papers from the FAA yourself.
The GCAA website has a lot of information and it's easy to navigate. You can contact them directly too.