The general rule (of course there are always exceptions) is that you owe tax to the state where you live and tax to the state(s) where you physically work.
In your case, because ME is not one of those states like NY that taxes telecommuters, if you never physically work in Maine, then you don't owe tax in Maine.
HOWEVER, note that if you ever go to Maine to visit the mother company as part of your job, then you need to keep track of the number of days and tell TurboTax, because then you will be liable to pay some tax in Maine.
If you pay any tax to Maine, then, yes, you will report it on the AZ return as "taxes paid to another state or country on Non-Arizona source income while a resident."
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