When he was ten, he was a finalist in a nationwide talent competition; winning for his piano rendition of Count Basie’s Jumping at the Woodside.
His debut album, Lift Off, featured Paul Rostock on bass, Bobby Durham on drums, and guest Joe Cohn on guitar.
This was a live recording on the S.S. Norway and included Jackie Williams on drums, Michael Moore on bass, and guest Gary Burton on vibes.
Jesse's 2002 album, Sylvan Treasure, featured guest saxophonists Phil Woods, Dave Liebman and Chris Potter, as well as Pat Dorian on trumpet, Bruce Cox on drums, and Frank Hauch on bass.
In 1998, Green took first place at the American Pianists Association's Fourth Biennial Jazz Piano Competition in Indianapolis, from judges Dr. Billy Taylor, Danilo Perez and Jim Turner.
In 2009, NPR reported that Green "spends most of his time teaching piano and trombone out of the home he shares [in Delaware Water Gap] with his wife and three daughters".
Green's trio (Bruce Cox on drums and Frank Hauck on bass) performed with Phil Woods at various concert venues in the tri-state area.