I am not a lawyer, but I do have the 2017 version of Traveler's Guide to the Firearm Laws of the Fifty States. According to the book, it would indicate that you will not be allowed to carry in CT with a VA non-resident permit. Their notes indicated that anyone who lacks a CT permit faces an unfriendly environment for gun carry. The book also states that CT has no general recognition of out-of-state carry permits.
Very simple answer, Connecticut does not honor any other states permits. You can not legally purchase or carry a gun in Connecticut with an out-of-state permit.