If you read The Thrill of the Chase, you’ll find a story about Forrest Fenn’s childhood Guernsey cow, Bessie. This story goes from page 28-32 in a chapter called Bessie and Me. There is also a photo of the family with Bessie.
As far as I can tell, it is correct to capitalize the G in the name of the cow breed. It is supposed to be named after the island in the UK where Guernseys were originally bred.
Some of you may know that a Guernsey cow is brown, or at least partly brown in color.
So some people might guess this has something to do with the home of Brown in the poem. A quick search reveals that there is a town named Guernsey in Wyoming; but the town is in the far eastern part of the state and way off the official chase map.
There is a Brown Cow dam in MT, but it also off the official map.
I haven’t found any promising instances of Bessie in any of the four states. I probably won’t look too hard.