We made the trip thru Monument Valley in 2014, by far our most ambitious TT trip. 4,700 miles and 3.5 weeks.
I researched which states along the way would honor my state concealed carry permit and which states would require other arrangements. However, I found absolutely no credible information on the rules on the Navajo reservation. Remember, they are a soverign nation and the federal rules do not apply.
In a fit of desperation and due diligence, I e-mailed the Utah Highway Patrol with my question and identifying information. Much to my surprise, an officer called my house the next morning.
He was very friendly and understood the problem well. He had worked in the southern area of Utah for 10 years and had tried many times to find their weapons rules without success. His advice was to keep it in a lockbox while on the reservation, and then when we moved out of the reservation we would be fine because he said Utah honors my state permit.
My point here is that it is NOT a dumb question. You'd better know what the rules are where, because every state is different. A unique example is NC where we love to camp - they honor my state permit, but their law requires you to proactively notify an officer that you are armed UPON FIRST CONTACT (you don't wait until he asks). I'll bet very few people know that.
Monument Valley was by far the highlight of our 3.5 week trip. We hired a Navajo guide for the tour thru the valley and he was very nice. We were fascinated by their way of life today and were able to discuss that with him. We were trying so hard to be polite and respectful that he finally started kidding us about it.
My only caution is to watch your fuel gauge. I've never been anywhere else where fueling opportunities were so scarce. We pulled our trailer up scenic highway 95 from Blanding to Hanksville and it was 144 miles between service stations and we didn't see 5 cars in that stretch. I had done my homework and knew about this, and we even went 5 miles out of our way to Blanding to fuel up before starting that trek (pulling with a gas pick-up). But the drive was gorgeous!
Tourism seems to be the only industry in that area and, while most of the scenic pull-offs have someone sitting there selling something, we never felt threatened. If you were interested you approached them, if you weren't interested they never tried to be pushy.
Don't miss this place! Just be sure you have everything you need before you get there.