Generally speaking, the children must reside in Arizona for at least six months before you can register the case here. There are two different types of "case transfer," and without additional information about your situation, I can't tell whether one or both is appropriate, so I am providing only general/basic information about both types.
1. You can register the foreign judgment with the Superior Court (if I'm not mistaken, Payson = Gila County). This involves obtaining an "exemplified" copy of the orders from Colorado and filing those orders with the court in Arizona. There is a filing fee to do so, and the other party must be given notice. Filing the judgment gives Arizona some limited jurisdiction over the CO orders.
2. You can file something with the court in CO, asking that court to relinquish jurisdiction of the child custody matters to the court in AZ (this would be done in conjunction with registering the foreign judgment). If the CO court will relinquish jurisdiction, then AZ would be the only court that can modify or enforce the custody/parenting time orders. If CO will not relinquish jurisdiction, you can still seek modification/enforcement through that court.
The process is a bit more complicated than I can adequately explain here and there areother jurisdiction issues to consider. You should speak with an attorney regarding the specific details of your situation.
** This is intended to be general information only and does not constitute legal advice or otherwise establish an attorney client relationship. **