The 37-year-old Greene, a defenseman, has spent all of his 14-year NHL career with the Devils and has been the team's captain the past five seasons. He will be an unrestricted free agent after this season.
He has two goals and nine assists and a plus-1 rating in 53 games this season. He has 49 goals and 197 assists in 923 games in his career.
"Andy has been a bedrock for the Devils franchise for the past 14 years and an exceptional leader in my time here," Devils executive vice president/interim general manager Tom Fitzgerald said in a statement. "I constantly communicated with Andy throughout this process to make sure he was comfortable with the decision, as it was not one I took lightly."
Quenneville, a 21-year-old defenseman, will report to Binghamton, the Devils' AHL affiliate.
After making the playoffs in 2017-18 but missing out last season, New Jersey traded for P.K. Subban, signed Wayne Simmonds and drafted Jack Hughes No. 1 overall. But the makeover hasn't changed the Devils' fortunes. Coach John Hynes lost his job Dec. 3, and the team waved the white flag two weeks later by trading Taylor Hall to Arizona. Other Devils are consistently mentioned in trade rumors.
The Islanders are currently third in the tough Metro Division but have gone only 5-3-2 in their past 10. They swept the Penguins out of the playoffs in the first round last season but then were swept by the Hurricanes in the second round.
The Islanders already play a defensive style under coach Barry Trotz, which has led to speculation that an offensive upgrade might be needed. But defenseman Adam Pelech is on injured reserve with an Achilles injury, so their first move ahead of the Feb. 24 trade deadline was on defense.