Why do you owe? Did NY change your returns or was the tax due not paid?
TurboTax guaranties 100% Accurate Calculations on a tax return. It does not guarantee if the user of the software omitted any type of income, entered an incorrect amount of income or deduction or answered a question in the software incorrectly.
See this TurboTax support FAQ for the Accurate Calculation guarantee - https://ttlc.intuit.com/turbotax-support/en-us/help-article/account-management/turbotax-100-accurate-calculation-guarantee/L30iuaxMC_US_en_US?uid=lod9jfh0
How to submit a claim under the guarantee - https://ttlc.intuit.com/turbotax-support/en-us/help-article/intuit-account-billing/submit-claim-turbotax-100-accurate-calculation-100/L88SYklBL_US_en_US?uid=lod9ka37
Documents needed for submission - https://ttlc.intuit.com/turbotax-support/en-us/help-article/intuit-product-orders/documents-need-submit-claim-turbotax-100-accurate/L8GE59ycB_US_en_US?uid=lod9l64m