Nope, that won't work. You can only get a licence in the jurisdiction (state, province, territory, island fortress, etc) in which you actually live, no exceptions. And even if you actually lived in CT, you still couldn't legally drive in NJ.
If you're driving on public roads inside any jurisdiction, you're subject to that jurisdiction's laws and rules. That's how it works, and no exceptions are allowed.
The minimum age for driving in New Jersey is 17 (I'm taking your word for this, because why would you lie), which means anyone under 17 is too young to legally drive in your state. If a kid from Connecticut is 16 years old and has a valid CT driver's licence, that kid still can't legally drive in New Jersey. That's because when you drive in New Jersey, you're subject to the New Jersey driving laws. The laws where you're from don't count for anything, only the laws where you are.