I own a house with someone I am not married to. She wants to buy property in another state. If I refinance on my own can we just do a quitclaim $$?

Yes, as long as that's acceptable to your new mortgage company, which probably means it's up to the title insurance company.

A mortgage loan has the house as collateral, but more specifically, it has the ownership interest in the house of whoever is on the mortgage, as its collateral.  This is why banks have to insist that anyone who is on the deed is on the mortgage.  In this case, the transaction has to clearly make you the only owner of the property.  A lawyer can help you make this all go smoothly.