It is true that there is no statute of limitation on bringing a paternity action in UT if there was never a presumed or declared father for you. But the problem is that this man is not in Utah. He is in NY. Under the relevant Utah statute providing for personal jurisdiction in these cases (UT Code section 78B-15-604, the UT court may exercise personal jurisdiction over him if any of the following circumstances listed in UT Code § 78B-14-201 exist:
(a) the individual is personally served with notice within this state;
(b) the individual submits to the jurisdiction of this state by consent in a record, by entering a general appearance, or by filing a responsive document having the effect of waiving any contest to personal jurisdiction;
(c) the individual resided with the child in this state;
(d) the individual resided in this state and provided prenatal expenses or support for the child;
(e) the child resides in this state as a result of the acts or directives of the individual;
(f) the individual engaged in sexual intercourse in this state and the child may have been conceived by that act of intercourse;
(g) the individual asserted parentage of a child in the putative father registry maintained in this state by the state registrar of vital records in the Department of Health pursuant to Title 78B, Chapter 6, Part 1, Utah Adoption Act; or
(h) there is any other basis consistent with the constitutions of this state and the United States for the exercise of personal jurisdiction.
So if you were conceived in UT then you may have a shot at it. At the moment, none of the other circumstances seem to apply here. You have to join your mother in the paternity action, by the way, and you would need her testimony to establish where the conception took place.
You also have the problem that you might lack standing notwithstanding the statute allowing a child to bring the paternity action at any time. The purpose of the courts is to provide persons a remedy for some wrong done to them or for relief that the law allows them. If there was some reason you needed to establish paternity in order to seek some kind of remedy or relief you would have standing. But seeking paternity just to satisfy your desire to know might not be good enough to meet standing. Talk to a UT family law attorney about this.
I can also anticipate that the court will order you to pay for all the testing, etc., if the court does allow the action since you are the one seeking the determination and you are doing it just to satisfy your desire to know, not to get support, etc.