Does filing a 1099-G qualify for the Free Edition?

Yes.   

Then why was I charged  $108? I had a simple return. I called last night and this morning to enquire about a refund for it but the lines were busy and I waited today as well but no one answered.

You could request a refund. See How do I request a refund for my TurboTax product?

 

If you used the Free version but it's charging you a fee, you may have entered a transaction that does not qualify for TurboTax Free file or you added some benefits.

 

Situations not covered in TurboTax Free Edition include:

• Itemized deductions
• Business or 1099-MISC income
• Stock sales
• Rental property income
• Credits, deductions, and income reported on schedules 1–3, such as the Student Loan Interest Deduction

 

I had none of those situations. Thank you for the link you provided. I will use it to enquire about a refund.