There are 4 US Post Offices in Greene County serving a total of 15,688 residents equating to 3,922 residents per post office. It's estimated that approximately 25,040 packages are handled across the 4 county post offices annually. Greene County is a county located in the U.S. state of Georgia. As of the 2010 census, the population was 15,994. The county seat is Greensboro. The county was created on February 3, 1786 and is named for Nathanael Greene, an American Revolutionary War major general."