Extremely unlikely.

Your friends are likely confused and thinking of the Tecaxic-Calixtlahuaca head, an artefact that was claimed by a Mexican researcher to have been found in the course of an excavation of an archeological site in the Toluca Valley in the 1930s.

The artefact does indeed resemble the head of a Roman statue (Severan dynasty), and lab analysis has confirmed it to be of ancient origin (1st - 2nd century CE). For now, the artefact is highly believed to be reliably of Roman origin.

However, the researcher who found the statue was known as a prankster, and it remains a possibility he himself placed the artefact in the excavation site as a joke to a colleague. Suspiciously, it was not until the 1960s that the researcher made his 'finding' public.

Even if the above is false, and this statue really did arrive in the Americas before European contact, it does not imply that the Romans or Old World peoples ever reached the New World and made contact with its inhabitants (aside from the Norse Vikings and their venture into Vinland). We know the Romans had outposts as far west as Lanzarote in the Atlantic, and it is not entirely impossible that a Roman shipwreck or its flotsam was carried by ocean currents to the shores of Mesoamerica, where it was found by the indigenous people there and who saw it as enough of a curio to treat it as a rare object and subsequently offered it up as a grave good.