I can't speak for the Aztects, but the Incas did take part in some revolts and uprisings against the Spanish rule both during and after the conquest.

The Inca Empire came to an end when the Spaniards executed Atahualpa, the last Sapa Inca (Inca Emperor) in 1533. Some years later they tried to re-establish the empire in the form of the later named Neo-Inca State, with capital in Vilcabamba (the original capital of the empire was in Cusco). This entity managed to challenge the Spanish rule for about 40 years, but eventually the Spaniards captured the monarch of the new kingdom/empire, Tupac Amaru, and executed him.

There were frequent revolts against the Spanish in all the formerly Inca-ruled regions (present-day Peru and Bolivia), although most of them were in protest of abuse and violence of the Spanish rule, as well as against the mita, and they were not necessarily attempts to overthrow the colonial rule.

In the mid-18th century there were massive indigenous uprisings led by Tupac Amaru II, a mestizo man who claimed to be a direct descendant of the last Inca emperor. Other uprisings include the one led by Tomás Katari in the 18th century and Tupac Katari later that same century, aided by a nephew of Tupac Amaru II. In these three cases they were trying to improve the rights of the indigenous peoples (mainly Quechua and Aymara) within the Spanish Empire, but their demands eventually included also liberation for the oppressed indigenous peoples. They were all captured and executed by the Spanish authorities, but they built the foundations of the 19th century independence wars in this part of the continent, and they are still remembered as heroic figures in Peru and Bolivia.