TurboTax cannot electronically file federal or state estimated tax payments.
See this TurboTax support FAQ for state estimated taxes - https://ttlc.intuit.com/turbotax-support/en-us/help-article/tax-payments/turbotax-calculate-estimate...
New York website for paying state estimated taxes online - https://www.tax.ny.gov/pay/ind/pay-estimated-tax.htm
