Many thanks VW and Niceguy
One thing's for sure, don't go to "Chinese Massage" behind LuLu Hypermarket.
I don't know about registered medical-type massage therapists, but most likely the better hotels e.g. Marriott, Four Seasons, and Ritz Carlton, have spas with good masseurs and masseusses.
Of course in Qatar only women massage women, and men massage men, so if you are used to good deep pressure from a male therapist at home, find a big woman here! ;o)
Oh, and there's a few massage places, some of which look suspiciously like Western style "massage" parlors, but aren't. These places offer Thai or Philipna style massages and nothing else; mostly for women customers. Look around Al Mirqab Street.
The short answer's 'yes' but quality and prices I don't know. The hospitals, both Government and private, have physio departments and there's a sprinkling of westerners with some training who work informally from home. If you're already in Qatar, ask around for the latter; or put an ad in one of the Western Social Club notice boards, or through the Doha Players Mailing list.