You do not file quarterly tax returns. You file one tax return at the end of the year.
If you are self-employed you probably have to make quarterly estimated tax payments. An estimated tax payment is just a payment. There is no quarterly tax return to file. You could use TurboTax to help you calculate how much you should pay, but you cannot make the payment with TurboTax.
When you say "filing quarterly", I am assuming you mean quarterly estimated taxes and not quarterly payroll tax returns for employers who have employee withholding.
We pay quarterly estimated taxes to avoid "underpayment penalties". To do that, a taxpayer can generally pay in 90% for his/her current year's total tax or 100% of the previous year's total taxes via Quarterly estimated taxes and/or withholding (110% if the previous year's Adjusted Gross Income is over $150,000). For Calif, if the taxpayer's income exceeds a million dollars, the 100 or 110% safe harbor is not available. So do watch out for any specific state tax law requirements.
The easiest and safest way to pay taxes is using ACH Debit. For the IRS, use www.irs.gov.