Helicopter flight instructor reported another helicopter on a converging resulted in taking immediate evasive action and a NMAC event.

Narrative

At or around XA:55 on Day 0; myself and a trainee for the company were en route from ZZZ1 with an intended landing at ZZZ for the purposes of tour training. Dual controls were installed in the aircraft; I was the pilot monitoring from the left seat and Person A was the pilot flying. We had just check out of the EWR class Bravo airspace over Bayonne piers and were transitioning across the upper bay to governors island when we experienced a near miss with another helicopter operated by Company X. Prior to the incident; EWR Tower had given traffic advisories for two ships inbound on the 'bridge' route. We picked up the traffic visually and reported that we would maintain visual separation with the traffic. EWR Tower switched us to the river CTAF and told us to squawk 1200; which we did. The trainee and I discussed making a radio call on the river to announce our position and intended landing zone. The student was on the controls and maintaining a constant heading and altitude concurrent with landing at ZZZ. As he maintained straight and level flight; I leaned down from the left seat to change radio frequencies to announce our position and then to contact ZZZ; but noticed an approaching aircraft out of the right side wind screen at the same altitude and on a course to intercept ours at a high rate of closure. I immediately grabbed the dual controls and initiated a climb to avoid almost certain contact with the approaching Company X aircraft. The student flying stated that he had not seen the aircraft approaching and was conducting a visual scan of the instruments at the time. The remainder of the flight was uneventful and landed at its intended landing at ZZZ. I called my chief pilot upon landing to notify him of the near miss. He contacted the landing pilots for Company X and stated that neither of them had been aware of the near miss. It is unknown which tail number or pilot flying for Company X was involved.