Yes - there is a branch:

Barclays Wealth
Qatar Financial Centre
Office 1002
10th Floor
Qatar Financial Centre Tower
West Bay
PO Box 15891
Doha
Qatar
+974 496 7515 / 7525

If you're an expat living / working in Qatar, you can open an international account with Barclays Wealth - you can open an account online or by phone. The website to look at is:

http://www.expat.barclays.com/qatar-bank-account/