In NYS, government pensions are completely excluded, while pensions and other retirement accounts from private employers or personal accounts get a $20,000 exclusion. So the first question is, did Turbotax incorrectly apply a full exclusion, or did you accidentally indicate this was a government pension?
You always owe your own correct tax. If Turbotax made a calculation error, Turbotax is supposed to cover penalties and interest. You must report the situation within 60 days of getting a tax notice, and you pay what you owe as soon as you can, because even if Turbotax accepts your claim, they won't cover extra interest or penalties if you delay paying. Here is how to file a claim,
https://ttlc.intuit.com/turbotax-support/en-us/help-article/account-management/turbotax-100-accurate...