The Dash 8 aircraft was tracking to the south after departure from Lockhart River. The aircraft had transmitted a taxiing call prior to departure. The call was acknowledged by the Flight Service operator but no traffic was given to the aircraft. As the aircraft was climbing through 4,500 ft the crew monitored a Lockhart River position report from a northbound aircraft, VH-PSQ. VH-PSQ was cruising at 4,000 ft. Both aircraft were in cloud and did not sight each other. The Flight Service operator reported that he had not recognised that the aircraft were conflicting traffic and as a result had neglected to pass the required information to each aircraft. At the time the operator had a high workload with other aircraft on his frequency, which was further increased by an aircraft that was out of communications.