How do I compute the taxes on a return that has income from 3 states?

Before you decide how the filing and income reporting need to be done in each of the states, you need to first consider the domicile of each of your clients. 

Domicile is not the same as statutory residency. Domicile is a common law concept based on intent and facts and circumstances.  Even if the husband was domicile in one state, his domicile may change with the relocation if supported by his intent and set of facts and circumstances.

That would not only determine whether a resident return may need to be filed for the state of domiciliary resident but also how the income needs to be divided and reported based on community property laws.  Note that both AZ and NM are community property states.

From the limited info you provided, the husband was not a statutory resident of AZ.  Why do you think he's a FY resident?  Not unless his domicile is in AZ.

If the husband was an NR of AZ but the couple wish to file MFJ for AZ, they'd file a 140NR and the husband will only be taxable on his AZ-source income.  How the various income would be divided between the couple, again, would depend on their domiciles.

At a high level, without considering your clients' domiciles and assuming the husband will file jointly with the wife for AZ, here's how you'd file the returns:

• AZ: MFJ 140NR with wife for FY; and
• NM: PY resident return for Jan

Credit will then be claimed on the resident state return for state tax paid to the source state on the income that has been subject to double taxation.  You should, nevertheless, bear in mind that division of income based on community property laws does not alter how the income is sourced.