This info can be updated on the CT law page:

If you change address, you have two days to notify the issuing authorities.

Change of Address
A person holding a state or local pistol permit is required to notify the issuing authority within two business days of any change of address. Persons holding state pistol permits may contact SLFU at (860) 685-8290 to update their address. The unit is open during normal business hours Monday through Friday of each week. A change of address may also be filed via E-mail at [email protected] or by mailing a letter to Department of Public Safety, Special Licensing and Firearms Unit, 1111 Country Club Road, Middletown, CT, 06457. Include your name, permit number, old address and new address.
DPS: Connecticut State Pistol Permit