Does turbo tax charge a quarterly filing fee for 1099-NEC?

You can prepare and print your vouchers when you file your return. Then, send them by mail with your payment on a quarterly basis. There is no extra charge for this.

If you’re at risk for an underpayment penalty next year, we'll automatically calculate quarterly estimated tax payments and prepare vouchers (Form 1040-ES) for you to print. Mail your payment along with the corresponding 1040-ES voucher to the IRS address listed on the voucher. Quarterly payment due dates are printed on each voucher.

Here are additional estimated tax payment options available at the IRS Payment website:

• IRS Direct Pay from your checking or savings account
• Pay online or by phone using the Electronic Federal Tax Payment System (EFTPS)
• Electronic Funds Withdrawal (EFW), only available during e-filing
• Same-day wire transfer from your bank
• Cash payment at a retail partner

You also have the option when you file your return to apply some or all of your refund to your estimated tax payments for the coming tax year.

You cannot actually  "file" your quarterly estimated payment thru TurboTax.  TurboTax can prepare the vouchers (IRS form 1049-ES). 

In your 2020 software, enter at:

Federal Taxes or Personal (H&B version)

-Other Tax Situations

  -Other Tax Forms

    -Form W-4 and Estimated Taxes - Click the Start or Update button

On the next screen answer No to the W-4 question

 If your goal is just to avoid the underpayment penalty, then paying 100% of the prior year tax liability is the “safe haven”

See ColeenD3's reply for how to actually file the vouchers.