Hii Shrishti!
Yes, a student who is studying in a private college or had passed out from private college can give a government exam whether it is for job such as railway, SSC CGL or for further studies. It would be better if u reframe the question and ask again including the details about the government examination which you are talking about. We will be able to serve you better if the question contains full details. Hope you understand.
Good luck!