Rio Grande
No, but the Colorado River watershed is primarily because of the Gila River.
The Gila is a tributary of the Colorado River and is 650 miles long. It is in the south-western states of New Mexico and Arizona. It has its source in western New Mexico.
From 1848 to 1853, the US-Mexico border in Arizona and New Mexico was the Gila River.
Gila River
The Rio Grande is the largest river of New Mexico, starting in the northern mountainous region of the state and winding southward through the middle until it becomes the border between Texas and Mexico. Other important rivers of New Mexico are the Canadian, Gila, Pecos and San Juan.
The five main rivers in New Mexico is the San Juan, Gila, Pecos, Rio Grande , and the Canadian river.
The Green River is a major river located in the western United States. The Gila River flows through New Mexico and Arizona. Goodhope River is in Alaska.
It was known as the Gadsden Purchase (1853), and it included present-day New Mexico and Arizona, south of the Gila River.
Southern New Mexico and Arizona, south of the Gila river. It is also known as the Gadsden Purchase (1853).
New Mexico
The West slope of New Mexico contributes to the Coolorado river watershed by way of the San Juan River which has its origin in Colorado, meanders through New Mexico and turns north to enter Utah and the Gila and Salt Rivers but the Colorado River itself does not enter New Mexico. The New Mexico Reservation mentioned in the question is the Navajo Nation and it does occupy lands on both sides of the Arizona and New Mexico Border as well as nortward iinto Utah and Colorado.
The Gadsden Purchase (1853), which included present-day southern New Mexico and Arizona, south of the Gila river.