As long as you are not prohibited from possessing firearms/handguns, you do not need a pistol permit when moving to Connecticut with legally owned handguns in your possession. It is recommended however that one obtains a non resident pistol permit before moving into the state then notifying the DEPP/SLFU to change the address on your pistol permit to your new Connecticut address.
Note: You cannot carry your handgun outside of your home/business, with limited exceptions, without a valid Connecticut Pistol Permit.
Connecticut General Statute Sec. 29-28.
Connecticut General Statute Sec. 29-35.
Connecticut General Statute Sec. 29-38. (b)