It depends.
If your gross income for the taxable year exceeds certain amounts as listed in the following link, you would need to file state income taxes within CT.
Connecticut Filing Requirement threshold
If you determine based upon the link above that you do have a filing requirement within CT, then you would be required to make estimated tax payment to the state if you anticipate your income tax due to be over $1,000 during the year. These payments will need to be paid on Form CT-1040ES and are generally due quarterly (April 15th, June 15th, September 15th, and January 15th).
Estimated payment dates and forms for CT nonresident
You would not need to make estimated tax payment to Massachusetts for the income earned in Connecticut. Instead, you will be able to claim a credit on your resident state tax return(MA) for any taxes paid within CT when you complete your MA resident state return. Just be sure to prepare your nonresident (CT) return within TurboTax first before you prepare your resident (MA) return to ensure the credits are calculated correctly.
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