Hello sir
Yes you can apply for 15% quota seats in private colleges.

(Meaning of 15% All India Quota Seats)
15% All India Quota Seats Means 15% Seats of All 460 Medical Colleges in India
(Including Private, Government,.all means all) will be filled via Centralised All India Counselling for the 15% Seats. Which means an Institution whether Government or Private will have to reserve 15 % of its seats which will be filled via All India NEET Counselling.
Rest 85% of the seats will be filled via State Counselling for the Students.
Thanku I hope this would be helpful