hi Shubham,

Yes you can take admission in government engineering college without JEE Mains too. But for that you need to give entrance exam test for that government engineering college. For eg. Jadavpur university for which you need WBJEE score not JEE Mains score. But keep one thing in mind that very very less good government engineering college take admission on the basis of some other entrance exam instead of JEE Mains. So, JEE Mains is a very important exam if you want to take admission in good government engineering college in India.

I hope this answer will help.
All the best for the future!!