Why did Turbotax questioned me to be able to file and receive a Michigan homestead property tax on the property I rent. MI wants that money back.

Did your daughter only pay rent for five months?  The Michigan Department of the Treasury website lists these qualifications.

You may qualify for a property tax credit if all of the following apply:

• You own or were contracted to pay rent and occupied a Michigan homestead for at least 6 months during the year on which property taxes and/or service fees were levied.  If you own your home, your taxable value was $143,000 or less (unless unoccupied farmland)
• Your total household resources were $63,000 or less (part year residents must annualize total household resources to determine if a credit reduction applies)
• Your total household resources do not consist solely of payments received from the Michigan Department of Health and Human Services

When I did her taxes I had only reported the rent she paid Jan - May (5 months - but still the software calculated the credit).  She did end renting a different place at the end of August, that I didn't report (didn't have the landlords info).  When she received the letter from the State, I told her to call and let them know - my mistake, I missed reporting the last 4 months.  The guy she spoke with said it's because she didn't make enough to qualify for this tax credit.  Now they want her to sign an Installment Agreement to pay back the $504.  I can't find anywhere where it states there's a minimum that needs to be made to qualify for that credit.  She doesn't receive any cash from the state (I do believe she was receiving food stamps for a bit).