The first time anyone can remember hearing it was in Liverpool at the start of the 1989-1990 season. The song was popular with Crewe Alexandra supporters before Manchester City fans started using it.
According to City historian Gary James, the start of Blue Moon being the City anthem was definitely on this occasiona in Liverpool. At the Anfield stadium, a group of City fans were delayed leaving, and as they left some of them began to sing this song. Nobody knows why, but several other people noticed them doing it as well. After that the club's fans were heard singing it at other venues too, and soon after it was adopted as their anthem.
What is not quite sure is when the chorus of blues first started doing the particular version that they often give today. However, you could contact the team's official website for further information.