No. You're responsible for your own taxes, not your employer.
I'm not from NY or familiar with their policy on first time underpayments. But their web site further says:
The underpayment penalty applies if your 2020 withholding and estimated tax payments are not at least the smaller of:
- 90% of the tax shown on your 2020 return,
- 100% of the tax shown on your 2019 return (110% of that amount if you are not a farmer or a fisherman and the New York adjusted gross income (NYAGI) (or net earnings from self-employment allocated to the MCTD) shown on that return is more than $150,000 ($75,000 if married filing separately for 2020)). You must have filed a return for 2019 and it must have been for a full 12-month year.
Reference: https://www.tax.ny.gov/pit/file/interest_and_penalties.htm#underpaying_your_estimated_tax
Whether they let first time filers off the hook, I don't know. Probably not. You could call them or just wait to see if they bill you.
