N.J. Const. art. V, § 1, ¶ 9

In the event of a vacancy in the office of Lieutenant Governor resulting from the death, resignation or removal of a Lieutenant Governor in office or the death of a Lieutenant Governor-elect or from any other cause, the Governor shall appoint a Lieutenant Governor within forty-five days of the occurrence of the vacancy to fill the unexpired term.

If a Lieutenant Governor becomes Governor, or in the event of simultaneous vacancies in the offices of Governor and Lieutenant Governor, a Governor and a Lieutenant Governor shall be elected to fill the unexpired terms of both offices at the next general election, unless the assumption of the office of Governor by the Lieutenant Governor, or the vacancies, as the case may be, occur within sixty days immediately preceding a general election, in which case they shall be elected at the second succeeding general election. No election to fill the unexpired terms shall be held in any year in which a Governor and Lieutenant Governor are to be elected for full terms. A Governor and Lieutenant Governor elected for unexpired terms shall assume their offices immediately upon their election.