I can suggest you two islamic centres as below, whom are conducting many Tamil islamic lecturers regularly at weekends.

Indo Qatar Islamic Council (IQIC)
(Besides FANAR building)
Telephone : 00974 4438678
e-mail : [email protected]
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QATAR INDIAN THOWHEED CENTRE (QITC)
(Near Najma Post office)
Telephone : 00974 4315863
e-mail : [email protected]