He will be replaced as chair by current managing director Martin Willis with Graeme Bunn becoming the new MD.
Gillham has spent more than 50 years working at Fleurets and said the time was right for him to take a back seat.
Gillham said: "I have worked at Fleurets for over 50 years so it will obviously be strange to wind down. However, the company's new chairman Martin Willis and I have worked together for over 30 years and, as managing director for the past 10 years; Martin is the ideal person to guide the board going forward.
"I shall keep my hand in by doing some rent reviews and arbitrations until I fully retire."
Challenge
Bunn, who joined the company in 2000, was the board's unanimous choice as the next managing director to take the business forward.
He said: "I am greatly looking forward to my new challenge. I take over the reins at a time when the business is in rude health and my colleagues and I look forward to driving Fleurets' reputation and experience across all leisure sectors.
"As a director for nine years, I have worked closely with Martin and am delighted that I have his, and the other directors', full support."