We own a house that we lived in until May 2024 then rented it out, would the mortgage interest paid be considered qualifying or non-qualifying mortgage interest?

It depends. This question is asked because you are entering this as mortgage interest in the deductions section of your return. Don't enter this here. if you enter any information here, remove it.

Instead, work on rental income and expenses in the rental estate section in wages and income section first. Here as you enter information about your rental, you will report your full mortgage interest here as an expense.  Then based on the information that you enter about your rental, TurboTax will allocate the correct amount to your mortgage expense for the rental expense and will allocate the remaining to Schedule A.