N.Y. Alt. County Gov't Law § 205
Vacancy in the offices of sheriff, coroner, county clerk, register, district attorney, and county treasurer shall be filled by the governor with the advice and consent of the senate if in session. When a vacancy shall occur, otherwise than by expiration of term, in the office of judge of the county court, surrogate, or judge of the family court outside the city of New York, it shall be filled for a full term at the next general election held not less than three months after such vacancy occurs and, until the vacancy shall be so filled, the governor by and with the consent of the senate, if the senate shall be in session, or, if the senate not be in session the governor may fill such vacancy by an appointment which shall continue until and including the last day of December next after the election at which the vacancy shall be filled. All other vacancies in an elective county office shall be filled by the board of supervisors.
N.Y. Alt. County Gov't Law § 205