There is some Government approved Institutions Like:

1. Hamad bin Khaled Al Thani Foundation to teach the Koran - Near Park and Shop - Ain Khalid

2. Good Doha Center for the memorization of the Holy Quran and its Sciences - Furousiya Street - Near Horse Club - Muaither area

3. Markaz Maryam Bint Imran for Holy quran learning - Abuhamour -

4. Sherifa Qasim center for the memorization of the Quran and its Sciences for Women..near Ainkhalid signal

5. Quran center for memorization - D Ring Road

6. Quran Hifz Center - Muaither

Etc...
You may search on google Map for nearby location of the centre...
Simply search Quran on google Map