No - If you are a permanent resident of NM, worked in CA but living temporarily in AZ, then you will need to file an CA nonresident state income tax return to report your wages while working in CA (which will be considered CA-soured income).
You will also need to file an NM resident state income tax return to report all income including your CA sourced wages. You will get a state tax credit on your NM resident state income tax return for any state taxes paid on your CA nonresident state income tax return for income that is being reported on both state returns.
You will have no AZ state income tax filing because you are only temporarily living in AZ and none of your income is from AZ sources.
Just follow the TurboTax guide when working on your states (remembering to do your nonresident state return first) and TurboTax will do all the calculations and credits to your resident states return.
Here is additional information about filing in multiple states (select "see more answer" to view the entire attachment)
https://ttlc.intuit.com/replies/3300797
