There is a full-time psychiatrist at the Doha Clinic. If you need someone who can prescribe drugs/medicines, you'll probably need to see her (someone with a medical license).
If you're simply looking for a therapist or someone to talk to about various issues, there are several freelancers in the expat community here. Most are western-trained or licensed, and some have different specialities (marriage therapy, substance abuse, work with children, etc.).
If you contact me privately, I can provide you with names and telephone numbers.