No, you will not enter him. The reason why is that your girlfriend is going to be claiming your child for all tax benefits. If you enter the child's information into TurboTax on your return, it will likely cause E-file rejections either for your return or for hers. Since you live together, only one of you may claim your child.
By the way, congratulations on your child's birth.
**Say "Thanks" by clicking the thumb icon in a post
**Mark the post that answers your question by clicking on "Mark as Best Answer"