The only person who knows John Amsterdam's secret about his immortality is Omar, who runs a bar in NYC. When John offered to make him a desk which would pay off his gambling debts, Omar remembered back when Thelonious Monk bought one of John's desks. It was after a midnight show at the Five Spot in 1957, when John Coltrane sat in.
Here's what Wikipedia had to say about those shows:
During the latter part of 1957, Coltrane worked with Thelonious Monk at New York City's Five Spot Cafe during a legendary six-month gig. Unfortunately, this association was not extensively documented, and the best-recorded evidence demonstrating the compatibility of Coltrane with Monk, a fund-raising concert at Carnegie Hall on November 29, 1957, was only discovered and issued in 2005. It was accidentally found in an unmarked box at the Library of Congress and issued by Blue Note, along with another of their recordings, The Complete 1957 Riverside Recordings. His extensive recordings as a sideman and as a leader for Prestige have a mixed reputation.
John Coltrane died on July 17, 1967, and he died hating John Amsterdam. In that first episode of 'New Amsterdam', we didn't learn why this was. But it may come up again, as the show will be revisiting John Amsterdam's memories of his 400 year past.
It could have been something recent, in the days leading up to his death, that really bleeped off Coltrane towards Amsterdam. One possibility could be that under whatever name Amsterdam was using back then, he could have been working as a cop in Newark, New Jersey. And maybe Coltrane was angry about the actions he took as a police officer during the Newark race riots which began on July 13.
Or maybe "Train" just didn't like the fact that John never made a desk for him.....
But that's all just speculation. We'll have to see if the reason is ever addressed on the show.