My girlfriend is trying to tell me that there is a paper that I signed when my son was born giving away all of my rights. And she believes with this paper, I would never get any custody. I am 99% sure that I didn't sign any paper like this and my name is on the birth certificate. Is there any way that the hospital would have given me a paper like this?

Not only wouldn't the hospital give you such a paper, but even if you signed such a paper it would have no legal standing. Only a court can terminate parental rights so just signing a paper would have no legal standing.

How old is the child and what is your current situation?

No such paper, she is lying or trying to con you.

Even if she typed up something and had you sign it, it has no value and is not valid, only the court can take away rights, and almost never do unless the custody parent is remarried and the new partner wants to adopt.

remember lying is the first thing, next expect her to make threats, like you going to jail, and that you will owe support from the day she got pregnant and more

You need an attorney, you need to file for visits or even joint custody

The child is 4 months. We currently just live together and unfortunately, even though we are a couple she insists on putting me through turmoil. She wanted to take my son to one of her friends and I was just curious where that was. She told me I didn't need to know and told me I couldn't take him anywhere either. So I told her I would go get custody so I could have him, and she then tried to tell me I signed this paper giving away all rights. I knew it couldn't be true, but I wanted to be 100% sure.

You are not in much of a relationship if she is telling you that you can not take the child anywhere.

If of course your name is not on the birth certificate, you will need to establish paternity

And you need to do this right away. You need to establish paternity and get joint legal custody so she can't keep holding this over your head.