Hello Guys,I am self-employed working in North Carolina. I have moved to New York from Nevada. Do I need to estimated taxes in both North Carolina and New York?

Probably just NC.  Although you have to file a NY return, NY gives you a credit for any tax paid to NC. 

If you used TT do to both your 2021 NC and NY returns.  See how much you owed. That should indicate whether you need to file estimated tax payments.

In each state software, there is a place for preparing estimate vouchers.  It's usually at a screen call "A few more things before we wrap up your state taxes" (or similar wording). 

For NC, You are required to pay estimated income tax if the tax shown due on your return, reduced by your North Carolina tax withheld and allowable tax credits, is $1,000 or more regardless of the amount of income you have that is not subject to withholding. 

For NY, you do not need to pay estimated tax if:

• You expect to owe less than $300 of New York State, $300 of New York City, and $300 of Yonkers income tax after deducting tax withheld and credits that you are entitled to claim.
• You expect your tax withheld during 2022 to be at least:
  • 90% of the tax shown on your 2022 return
  • 100% of the tax shown on your 2021 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 2022)). You must have filed a return for 2021 and it must have been for a full 12-month year.