Graham Moxham had been with both Preston and Bournemouth as a youngster before he joined Exeter City from Bideford in 1975. Making his Football League debut at the age of twenty-six, he played half-a-dozen times in the league before moving on to Barnstaple Town.

Graham started his career as an apprentice at Preston North End and later became a full professional  with Bournemouth in July 1966. Not making the first-team at either club, Graham played non-league football until joining Exeter City from Bideford in July 1975.

At the age of 26 Graham finally made his senior bow as a substitute in the League Cup game at Torquay United on 10 September 1975. Following this up with another substitute appearance in the league game against Cambridge United three days later, he went on to start four league games - as well as the FA Cup tie against Cardiff City - before his final appearance against Doncaster Rovers at the end of November. 

Leaving St James’ Park later that season, Graham continued his non-league career with Barnstaple Town.