Mom always tells me that if anyone goes to nursing home that they put them on Medicaid. Is this true?

She fears a NH which I sympathize with. In her parents/grandparents time NHs were not so great. State intervention has helped. Medicaid has a stigma that the resident on it will not get the same quality care as the person private paying. Which according to my daughter is not true. Staff are not aware of who is or who isn't on Medicaid. All residents are treated the same.

Your Moms current health insurance (Medicare plus supplemental) have nothing much to do with paying for a nursing home. If she has enough savings to pay for the $10K or so a month that it costs for a nursing home, she will not need to go apply for Medicaid. If (or when) she doesn’t have the money to self pay, she will apply for Medicaid to pay for the NH, and would probably stay with her current Medicare health insurances.
Why is she afraid of being on Medicaid?

People go on Medicaid in a nursing home when they have self paid and have run out of money. This doesn't mean that everyone is self pay from the beginning. Not all nursing homes accept Medicaid. Some require that resident self pay for a period of time before they will accept Medicaid and provide a Medicaid bed.