Can a nursing home legally transfer a patient out without notifying the family or financial POA?

If there is no Medical POA, then the patient signs for the transfer. Seems a bit shortsighted, especially if the patient has memory loss of any kind. In fact, the patient could even sign out and go home with no one knowing.

What does the signed agreement say?

The nursing home can do things that ensure the safety and care of the people living there. I don't know about your friend's situation, but if the facility could not adequately care for the patient, or the patient was a threat to self or others, then they would be negligent in not moving them to a more appropriate facility. They would also be able to do it if the family fell short on their side of the contract and did not respond to requests. You would have to provide a lot more information before your question could be answered well.

I believe technically, I she hasn't been declared incompetent, they can do so.

The nursing home was in transferred her to another nursing home without contacting anyone

Transfer her out where? Are you talking eviction or going back home?