There are quite a few anti-depressants which are banned in Qatar, but some SSRI's should be available over the counter. Many people recommend the Khulood Pharmacy at the Burger King roundabout in Doha (if you aren't actually there yet, roundabouts are often just named for something nearby).
It would be a good idea to go there first because it may save what could potentially be a lengthy process. I've heard some conflicting reports about this, but you should be able to get antidepressants prescribed by a GP in Qatar. For some classes of medication, it may be necessary for the GP to refer you to a psychiatrist.
The short, provisional answer is yes.
To alert yourself to how the healthcare system in Qatar works, check out our page.
Best of luck
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