My daughter is 3 months old and the father is between two different guys, neither of them will get a DNA test and I am currently in a relationship. Without a father's name on the birth certificate can my current boyfriend put his name on the birth certificate without a legal process?

That is considered fraud and is illegal, your current boyfriend has perhaps made hisself responsible for 18 years of child support if you two break up latter.

And you don't ask them to take a DNA test, you go though court and get a court order to force them to do a DNA test, you prove who the father is, and collect child support.