My mother needs Medicaid to be accepted into a nursing home.

Most facilities do offer pastoral care. My MIL lived in a corporately owned facility that had Mass weekly and regular visits from a priest or nun. Just ask if the facility has these services. Given the high population of Catholics in NH, I think this will be easy.

My parents have lived in New Hampshire since 1979. My father moved my mother 2 hours away because there is a good nursing home in Maine. He would like her to be accepted into a local nursing home that is Catholic but they said she has to qualify for medicaid before they will accept her. My parents have enough money to pay for care for a few years. When I was asked I was concerned that moving her to a different state could be a problem with medicaid. Her home address is still New Hampshire. Maine is supposed to be a temporary stay. She doesn't have medicaid yet because she has too much money to qualify but it is needed to be accepted in the nursing home we hope she will be accepted into.

If she has Medicaid in Maine, it stops as soon as she leaves Maine. In the new state she has to establish residence and apply for New Hampshire Medicaid.
If your father can afford to privately pay for a Nursing Home, any facility would be glad to have her. Your father probably would prefer to keep her close to him, though and I think you should respect that.

Laura, each State has their own rules, regulations, and programs. Hopefully your Mom can qualify and be able to get into the nursing home of choice.