You would need to check with the court listed on your speeding ticket that you received while on the Navajo Nation land in Arizona to see if they share information with the Arizona courts who in turn would inform the California Department of Motor Vehicles either about the conviction of this ticket or none compliance (failure to appear) if you do not pay it.
We have read and heard from others in the past that citations received on Native American tribal land / reservations are sometimes taken care of in State courts and not tribal courts. If this is the case then it likely would be dealt with as if the ticket was issued by the state patrol instead of the Navajo Nation. In that case then Arizona would notify California of the ticket if you paid it and thus were convicted. If you choose instead to not pay the ticket then likely your driving privileges to operate a vehicle in Arizona and the Navajo Nation would be suspended.
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[Let CarInsurance.com help you find affordable auto insurance now.] We would again advise that you contact the court listed on the speeding citation from the Navajo Nation to find out more information on who they inform about the violation, whether you pay or not. Also the Arizona Motor Vehicle Division (MVD) should be able to give you information on if they receive ticket information from tribal lands and how they deal with convictions or those that do not comply or respond to a ticket. | 
