My mother is on medicare and ohio medicaid and is in a nursing facility. What happens if she gets an inheritance?

Yes - that's possible.  If your mother receives an inheritance, she may have to pay back medicaid, and then spend what is left of the money for her care, and then when that runs out go back on medicaid.  It may be better if she does not accept the inheritance.  You should discuss this issue with an Ohio elder law attorney.