If you are the nonresident spouse of a service member stationed in New Mexico, you do not pay New Mexico income tax on the wages you earn in the state.
There is nothing to apply for with a state office. Claim the exemption when you file, and give your New Mexico employer your out-of-state residency information so state tax is not withheld from your wages.
Your spouse’s military orders assigning them to New Mexico, and proof that you and your spouse are residents of another state.