Actually, I just called the Navajo Nation in WIndow Rock and was connected to their police division (their version of DPS) and was told the following:
If you are a card carrying Tribal Member (Navajo Nation Member), Tribal Law applies to you. You are permitted to own a gun but no open or concealed carry is permitted, period. The Navajo Nation does not have a weapons permit system.
If you are a US Citizen, Arizona Law applies to you... just the same as if you were in Phoenix, Flagstaff, Payson, Prescott or anywhere else in the State. Your CCW is honored, concealed carry is OK, ect...
NOTE: If you are an Open Carry person, You may get stopped and asked to put your sidearm away and/or leave the reservation. Carrying a weapon, of any type, is considered by the Tribal Elders as a sign of agression and considered very disrespectfull.