Welcome to Doha.

You don't need a license. I recommend you adopt a dog from Qatar Animal Welfare Society (QAWS), they have dogs that need a master :

http://www.qaws.org/index2.php

https://www.facebook.com/qaws.org

For import/export, you can try:

Qatar Pet Relocators: http://www.qatarpetrelocators.com/index2.php

Qatar Veterinary Center: http://www.qatarvet.name.qa/QVC/HOME.html

Qatar Veterinary Surgery: http://dohavets.com/index.html

Qatar Pet Travel: http://www.qatarpettravel.com/importexport.html

Qatar Pet Relocators seem to be the most popular.