You do likely owe child support, but you don't have to do as she says. She can't have it both ways, have you pay and decide whether you can see your own child or not. I would probably petition for custody in consultation with a Family Law attorney.
You should immediately file a petition for visitation and custody. You will have to explain why you have not done this prior. You may need to demonstrate what efforts you made to locate your child. You have a right to a relationship with your son and should now work towards this. Given the circumstances it may be a long road and you may meet resistance from your child. But to not try would be more detrimental. Consult local counsel.
The above answer does not constitute an attorney client relationship and/ or retention of counsel. This answer is based upon the facts presented and may change if additional information is provided. The rules of the Bar for New York State may require me to advise that this could be construed as attorney advertisement.
If she has held another man out as your child's father, you likely have a strong argument under a doctrine called "estoppel" to prevent her from collecting child support from you. However, if you want visitation and parental rights with your child, you must go to court to seek those rights and would then also be expected to pay child support.
I also agree that estoppel may apply here if another man has held the child out as his own. However, if your ex files for support and her paramour signs an affidavit of no access, the Court can in fact order you to submit to a paternity test. Nevertheless, you should file a petition for visitation if you want to establish a relationship with the child. Unfortunately, it would be quite a feat to beat her in a support proceeding and wind up with a parenting schedule at the same time.
The opinion herein does not constitute legal representation in any way or establish an attorney-client relatioship.