An account of how Cortez received its name is provided in papers given to the Montezuma County Historical Society several years ago.
Although its source (and accuracy) remains uncertain, the article follows:
CORTEZ: The town of Cortez is named for Hernando Cortez, renegade Spaniard who disobeyed Diego Velasque and in 1519 sailed to Mexico and after 18 months of bloodshed conquered Mexico. In the battles with the Aztecs 250,000 Indians were killed or died of starvation. The first chief whom Cortez captured, Montezuma, died in prison. His successor, Guatemoc, surrendered August 13, 1521. Cortez never got into what was to become Colorado. He died in 1547 and his body was moved many times, finally being placed in the church of the Hospital of Jesus Nazarene in Mexico City. It was concealed in the wall from 1823 until 1946 to prevent desecration due to anti-Spanish feelings.
Source: Great Sage Plain to Timberline: Our Pioneer History, Volume 1.