Yes you can get the college through state quota in bihar and the chances of getting the govt. college are high for you as govt. medical colleges require 500+ marks in NEET. The cutoff marks for the previous year's were 533 with the rank 11253. So you go through the counselling rounds and get your admission. A link is provided for the further info about the govt. colleges in bihar and their rank.

https://medicine.careers360.com/articles/neet-cutoff-bihar

Dear Candidate

See as per your Marks-537 it is difficult to get a Government Medical Seat even if you have a state quota and another quota if applicable. You should have maintained a score above 550 to be on the safer side

At this stage, I will suggest you must use this link to get a fair idea about the colleges both Private and Government you can get as per the score you have received and then plan accordingly.

https://medicine.careers360.com/neet-college-predictor

https://medicine.careers360.com/articles/neet-cutoff-bihar