Your home state of Connecticut and the state of New York are both members of the Drivers License Compact (DLC) which allows states to exchange information regarding violations received out of state back to the state in which a person is licensed.
The CT Department of Motor Vehicles typically assigned points to a person's driving history if they are convicted of a violation but not if you make a payment in full and dispose of your case. The DMV does not state whether points are different if the violation is from out of state.
Check with the Connecticut Driver Services Division to find out for certain if points are assigned, and if so how many, for out of state moving violations. The CT DMV site says to call for information on this at 860-263-5720, the Operators License phone center.
Comment Update:
You can read more about Statute 14-111g on Connecticut's statute page, but its history reads that it made the retraining program applicable to all licensed motor vehicle operators by deleting the provision limiting such program to operators under eighteen years of age and the provision authorizing the commissioner to suspend the operator's license of an operator who does not successfully complete the program until the operator attains the age of eighteen years, effective July 1, 1995; P.A. 98-182 amended Subsec. (a) by creating a suspension violation, altering the requirements for attending a retraining program for persons twenty-four years of age or younger and persons over twenty-four years of age, expanding the retraining program to include the development of alternative attitudes for those contributing to aggressive behavior and an emphasis on safe driving behavior and added a new Subsec. (b) allowing the commissioner to retain a portion of the fees collected for the cost of implementing the retraining program, effective July 1, 1998; May 9 Sp. Sess. P.A. 02-1 amended Subsec. (a) to delete reference to Sec. 14-227a(b), effective July 1, 2002; P.A. 03-278 made technical changes in Subsec. (a), effective July 9, 2003.