Montana's three biggest rivalries are:

Montana vs Montana State (Brawl of the Wild)
Montana vs Idaho (Battle for the Little Brown Stein)
Montana vs Eastern Washington

There is no official name, or trophy, for the Montana/Eastern Washington rivalry. However, being the two most dominant Big Sky teams in the 2000s and 2010s, and only 213 miles of highway separating Missoula and Cheney, this has become a large rivarly in the Northwest.