A hotel union that already wields considerable influence in New York City politics is now aiming to beef up its presence in New Jersey. The Hotel and Gaming Trades Council opened its first office in the state — in Montclair — on Wednesday, the same day workers it represents at 10 hotels in New Jersey voted to ratify contracts. Employees at 57 hotels in the state are now connected to the union, which represents about 40,000 hotel and gaming industry employees in New York and New Jersey, about 6,000 of which live and work in the Garden State. “We’ve been in New Jersey now for about a decade,” Ri…