There are some places where you can learn Arabic:
Fanar/QICC
Location: In front of souq Waqif

Qatar Guest Center
Location: Kulaib
Contact No. : +974 - 486 - 2390 / 7
Website : www.qatarguestcenter.com

Language Institute
Location: Salata Jadeed (Near Indonesian Embassy)
Fee: 200QR/term
Contact No: 465760

Markaz Maryam (Maryam Center) --> Women Only
Lokation: Mamoura
Fee: Free
Contact No: 5291785