Michael Conway, son of founder Frank Conway, will step up to become chairman.
FM Conway started from very humble beginnings as a road mending business with just three men and a lorry.
Michael transformed it over the past 39 years into a major civil engineering business employing over 1500 people with revenue over £325m.
He said: “My father had a dream in 1961, and by 1981 that dream had been realised so I was passed the baton to take the business to the next stage.
“Today, it gives me great pleasure to pass that baton on once again.
“FM Conway will now enjoy a fresh pair of eyes and a new set of hands to navigate the business into it’s 60th year and beyond.
Green said: “I am both delighted and honoured to take on the prestigious role of CEO.
“I have long admired the business and am keen to ensure that the strong family values and people remain at the heart of what we do.
“Our unique self-delivery model truly stands us apart from our competitors and I am excited to explore this even further as we look at new technologies to bring greater levels of service to our customers.”