You do not file quarterly tax returns. Only one tax return is filed each tax year reporting your total income and expenses for the year if you are a self-employed independent contractor. Schedule C is used to report your SE income and expenses which is included and part of your personal tax return, Form 1040.
Using the TurboTax online editions you will need the Self-Employed edition for your self-employment.
You can pay estimated taxes during the tax year using Form 1040-ES and the worksheets available in the instructions for the form - https://www.irs.gov/pub/irs-pdf/f1040es.pdf
Estimated tax payments can also be paid using the IRS payments website - https://www.irs.gov/payments
You can use TurboTax to assist in calculating the estimated taxes using the procedure in this TurboTax support FAQ - https://ttlc.intuit.com/community/tax-payments/help/can-turbotax-calculate-next-year-s-federal-estim...
