This is, in general, true. According to the custom of the time natives changed their natives and adopt both a name and surname that were "Christian". Both were usually chosen by their "sponsor" that in some cases was the priest performing the ritual or a Spaniard that was acting as a sponsor that usually picked the named in honor of some relative, an important figure of the time, etc. Some of the priest did indeed used their very own last names. That being said, is important to remember that no every ethnic group (I wouldn't say "none had" because I'm not sure, but no group pops at the top of my head) had the idea of a surname in their culture. Sure, they did have the idea of clans, families, blood line, etc. but not the idea of a surname in the way the Spaniards and other Europeans had.

That's why is important to see which "surnames" remained. I'll give you a couple of examples: Moctezuma was the last emperor of the Aztec Empire. He's name was Moctezuma ("He who frowns like a lord"). That was his name, not his surname. As a matter of fact, in the Nahua codex that remain we have no mention of a last name, just the addition of "Xocoyotzin" ("The Young") to differentiate him from an oldest Moctezuma that reign before him. He was not baptized before he died/was murdered so that is the only name he had: Moctezuma Xocoyotzin. However, his many daughters and sons were baptized. I'm going to focus on one that was recognized his legitimate heir. She was around 10 years old when her father died and her name was Tecuichpoch Ixcaxochitzin. Tecuichpoch is actually more like a title: The virgin/young daughter of the Lord. Ixcaxochitzin, on the other side, would translate as "White flower" or "The flower of the corn", (-tzin being revential suffix used for nobility) so she was White Flower, the young daughter of the Lord. As you can see, a surname is not mentioned. When his father died (according to Cortes, in his Conquista de Mejico) in his last moments begged Cortés to protect his three daughters for they were very dear to him, specially the oldest (Ixcaxochtzin) and begged him to convert them to Catholicism and to end their “idolatry” (again, this is Cortes so take his words with a ton of salt). So, Cortes had procured himself an heir for the late emperor, his 10-year-old daughter. So, he baptized and gave her the name Isabel (probably to honor Queen Isabel of Spain), the honorific treatment of “doña” (lady) and the surname Moctezuma. So, her full name was Doña Isabel Moctezuma. Cortés used her father’s name as her surname in an effort to gain legitimacy for her and to remind people that she was nobility. This was not the only case, as in general the Spaniards of New Spain (current day México) tried to respect the Aztec nobility in an effort to draw parallels with Continental Spain and to create the image of a certain continuity. As a matter of fact, when Doña Isabel had issue with Cortés the daughter was named Doña Leonor Cortés y Moctezuma. This is a huge contrast with Malinche, Malinalli (Queer Vine), Cortés translator and a piece of utter most importance to the Conquest (for a time Cortés was named as Malintzin, The Lord of Malin). She was noble-turned-slave that spoke both Nahua and Mayan and later Spanish. Even with this importance (recognized by Cortes and Díaz del Castillo) she was not an important figure in a realpolitik way. She was baptized doña Marina, but there is no record of a surname. And the son she had with Cortés (don Martín Cortés) is usually mentioned with just one surname (his fathers) and, in the rare cases he in mentioned by his two surnames, his mother’s surname is Malintzin. So, in this case she was probably given some random surname that no one bother to register because she was just a slave translator.

This is not, by far, the only case. In Perú, many members of the royal family were baptized with the surname “Yupanqui”, the name of an emperor.

So, to answer: yes, that is true, however is also worth mentioning that not all the natives had the concept of a surname so there wasn’t a native surname to lose in the first place.