Your cousin can not drive in New York State , even if they have a permit or license issued by another state.
does my learner permit allow me to drive in New Jersey?
You might be able to drive outside New York State with your NY State junior learner permit or junior driver license if it is allowed by the laws of the other state.
what about the other way around, can someone who is 17 and has a learner permit from New Jersey drive in New York?