Under New Jersey law (New Jersey Statutes Secton 26:8-28),
Quoting New Jersey Paternity Law
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Within five days after each birth, there shall be filed with the local registrar of the district in which the birth occurred a certificate of the birth filled out with durable black or blue ink in a legible manner. The name of the father shall be included on the record of birth of the child of unmarried parents only if the father and mother have signed a voluntary acknowledgment of paternity; or a court or an administrative agency of competent jurisdiction has issued an adjudication of paternity.
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So you can't simply add the putative (presumed) father's name to the birth certificate without his consent.
Your other option is to commence a paternity action, which is filed with your county court. If the putative father is properly served and doesn't answer the complaint, he would be declared the child's father by default. If he does appear, DNA testing would be performed to confirm parentage. If adjudicated the father, he would have the duty to pay child support, and the right to request parenting time.