Is it Koranic or Qur'anic schools? Do I need to use a capital?
Both of the spellings you suggest are correct. Quranic is another spelling variant that is also acceptable. Although Qur'anic and Quranic are closer to the Arabic, they have not replaced Koranic (derived from the proper noun Koran) in Canadian English usage.
For texts written for a North American audience, we suggest that you use the adjective Koranic, with a capital K, since this is the better-known spelling.