He is understood to be 33, married with a young child, Otago Cricket Operations manager Tim O'Sullivan said.
He is of Indian ethnicity and played in a team made up predominantly of Indian players.
The club will be required to complete an incident report but New Zealand Cricket's manager of public affairs Richard Boock said there was no indication it was a sports mishap.
"Early indications are that the player has had a heart attack. It is tragic. Everyone is shattered, both at the club and Otago Cricket. Our hearts go out to his family."
St John Ambulance confirmed that they attended the incident with two vehicles.
Green Island is a five-time National Club Champion and current Dunedin champion.
It was established in 1930 and its website describes it as having a warm, family vibe.
The club was home to several Black Caps including Glenn Turner, Hamish Rutherford and Brian McKechnie, who faced the infamous underarm bowl in a one-day international against Australia in 1981.