No. They called it 'New Spain'.
Viceroyalty of the New Spain
Vice royalty of the New Spain.
The colonial Mexico of Spain was known as New Spain or the Viceroyalty of New Spain. The colony was ruled by the Spanish King and he appointed a viceroy, or very high minister, to assist in the running of the colony.
The Spanish called the new colony they established on the site of Tenochtitlan "Mexico City" after conquering the Aztec Empire in the early 16th century.
Taos
No, there is no placed called Mexico, New Mexico.
because they found wealth in Mexico, so hoping they found it there they named it New Mexico
That is a colony.
Taos, New Mexico
The Pilgrims called their new colony Plymouth.
Viceroyalty of the New Spain