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
NY is a bit special for those....most states don't allow you to set up those estimated payments thru the software.
Several years ago, the NY software, it did allow you to "arrange" the setup of NY quarterly estimated tax payments (but that was back for 2017, don't know if they still do).
However, this would be done at the time you e-filed the original NY tax return.....and I kind-of doubt the software allows it to be done separately later. Even if done thru the software at the time you filed your original tax return....TTX would only transmit the data to NY, and then NY would actually do the debit according to how you set it up.
At this time of year....Just paying it thru the NY website (using the NY weblink @DoninGA indicated) is probably easiest and avoids any potential software complications and finger-pointing as to who is responsible for getting it done.
Here's some pictures from several years ago where the NY payments were set up with the desktop software at time of filing:
1) Part of the NY Information Worksheet in Forms Mode
2) Section of the Electronic Filing instructions sheet when filed (different $$ amounts though)