No, all income must be filed with the return for the year in which you received the income. 2020 income goes only on a 2020 tax return.
If you have already filed your return, then wait to be sure it is accepted by the IRS. If for some reason it is rejected, then you can include the 1099-G in your return before you resubmit it. Otherwise, if your return is accepted, then you will need to file an amended return in order to include the 1099-G with your 2020 return.
Keep the following TurboTax article handy for reference. Later in the tax season when the amended return form and instructions are released you will find step by step instructions by clicking the links in the article.
How to amend (change or correct) a return you already filed