The incident happened at Ringsend, near Coleraine in Northern Ireland, on Saturday afternoon.
He has been named locally as Hugh Henry, a father of two.
A spokesperson from the Health and Safety Executive (HSE) said they were aware of the tragic incident and are making enquiries.
"The Health and Safety Executive extends its deepest sympathies to the man’s family at this difficult time," he said.
A NI Ambulance Service spokesman said: "On their arrival the crews found that a male patient, in his late 50s, had sustained fatal injuries in the incident.".
"He was pronounced dead at the scene. Our thoughts and prayers are with the family and friends of the deceased."
A family notice described him as the ’beloved husband of Margaret and loving father of Hugh and Maria’.
SDLP East Londonderry MLA John Dallat said: "I don’t know exactly how it happened yet. Hugh is a native of Ringsend and is very well known in the community and highly respected as a farmer," he said.
"The wider community will be in shock and my sympathy goes to him and his family.
"I would call for all the support they can get at this very difficult time.
"Farming by its very nature is a very dangerous occupation; really it’s only at times like this that we realise that.
"It’s a hard industry to be involved in and even with the best precautions there’s always an element of risk.
"There has been a number of previous fatalities involving quads but now is not the time to speculate about what happened."
Sinn Féin MLA Caoimhe Archibald said: "This death has shocked the local community and I send my sincere sympathies to the family and friends of Hugh Henry at this difficult time"