Greenock is a census-designated place in Elizabeth Township, Allegheny County, Pennsylvania, United States. As of the 2020 US Census the population was 1,989. The community was named after Gre…Greenock is a census-designated place in Elizabeth Township, Allegheny County, Pennsylvania, United States. As of the 2020 US Census the population was 1,989. The community was named after Greenock, Scotland.