Quarterly payments

You cannot pay the quarterly estimated tax payments directly through TurboTax.

TurboTax has a feature where you can calculate your estimated tax payments.  Those payments will be entered on a Form 1040-ES that you can print and mail to the IRS.

See this TurboTax support FAQ for calculating next years estimated tax payments - https://ttlc.intuit.com/community/tax-payments/help/can-turbotax-calculate-next-year-s-federal-estimated-taxes/00/26437

See this TurboTax support FAQ for making estimated tax payments - https://ttlc.intuit.com/community/tax-payments/help/how-do-i-make-estimated-tax-payments/00/25875

Hi Lashford!
 
You are correct, you can not make your quarterly estimated tax payments through TurboTax.   
According to the IRS website:  https://www.irs.gov/payments , you have a few choices to make payments (including estimated tax payments).
 The options are:  
1. You can print the 1040 ES vouchers and mail a check or money order to the IRS.   

2. You can go to IRS Direct Pay (known as Electronic Federal Tax Payment System) and make the payment electronically debiting your bank account.  When making the online payment, make sure you choose ESTIMATED TAX as the reason for payment and apply it to 1040ES with the appropriate TAX PERIOD.

3. Pay with cash is an option in some situations.  Please check IRS payment page, https://www.irs.gov/payments/pay-your-taxes-with-cash, for assistance with locating current retail partners as well as any instructions with cash payment limitations and phone numbers of IRS centers that accept cash payments. 

4. Electronic funds withdrawal is a fourth option, but only when it is combined with e-filing a tax return.  

I am not sure of which state you are currently located.  If you have state estimated tax payments, please note that most State Department of Revenues have online payment options on their respective website.