Joe Graham played for Exeter City during 1921/22. He joined the club from Stockport County, where he played either side of the war, and later played for New Brighton.

After playing for Tyneside club Wallsend Park Villa, Joe signed for Stockport County in the summer of 1911 and was to remain at Edgeley Park either side of the Great War until the end of the 1920/21 season. Rarely a regular during his time with the club, he had scored three goals in 86 league matches for Stockport by the time he joined Exeter City in the summer of 1921.

Making his debut at Portsmouth in the seventh league game of the season on 17 September 1921, Joe had a good run of twelve games before his final appearance at home to Watford on Christmas Eve. 

Leaving St James Park at the end of the season, Joe signed for New Brighton before later turning out for Ashton National.