I live in Arizona, but child lives in another state. Nothing has ever been filed in Court. Can I file in Arizona to get custody?

It depends. For Arizona to have jurisdiction over custody, it has to be the home state of the child. That means Arizona has to be the state where the child most recently resided for six consecutive months. If Arizona is not that state, then Arizona cannot make a custody order, and you will need to file in the state where your child has most recently resided for six consecutive months.