At the very end of your New York state return, you should see a screen that says Additional Tax Situations. On this screen, there is an option to Prepare Estimated Tax Vouchers for 2019 (Form IT-2105). Please see attached screenshot for reference. 

I am not asking about Estimated Tax Vouchers.  I am asking about "automatic payment of quarterly estimated taxes for the state".
If in fact the option to Prepare Estimated Tax Vouchers, does give the opportunity to set up automatic payment of quarterly estimated taxes for the state for 2019, then how do I do that once the return has been filed?

I have the same question!