tim in the comments pointed to the snopes article but I'll quote it here.
This particular claim is mostly false. The snopes article addresses a list of claims from an image including this one.
For one, it wasn't a club, segregated or otherwise when he bought it. He simply didn't make it segregated when he made it a club and wasn't asked to by the city.
But Trump didn't literally buy a "segregated club," as when he purchased Mar-a-Lago in 1985 it was a private estate and not a club. It wasn't until the 1990s that Trump sought to convert the property into a private club, whereupon the town council imposed a series of restrictions upon his plans. Those restrictions didn't require that Trump operate a segregated club; rather, Trump battled those restrictions, in part, by making the point that it was unseemly for the town to impose greater restrictions upon his plans to open an integrated club than they did upon existing segregated clubs:
The town council, seeing Trump as an ostentatious outsider, handed him a list of restrictions as he sought to transform the property in the 1990s. Membership, traffic, party attendance, even photography — all would be strictly limited.
But Trump undercut his adversaries with a searing attack, claiming that local officials seemed to accept the established private clubs in town that had excluded Jews and blacks while imposing tough rules on his inclusive one.
Trump's lawyer sent every member of the town council copies of two classic movies about discrimination: A Gentleman's Agreement, about a journalist who pretends to be Jewish to expose anti-Semitism, and Guess Who's Coming to Dinner about a white couple's reaction to their daughter bringing home a black fiance.
The move infuriated council members, who said it was a distraction from their concerns that Trump's club would spoil a quiet street. But, in time, Trump got most of the restrictions lifted.
"He won in the court of public opinion," said Jack McDonald, who was a council member at the time and who went on to be mayor and to join Mar-a-Lago.
Town council member Allen Wyett opined that Trump's nondiscriminatory policy at the club was more of a business decision than a social one:
Wyett, who is Jewish, said he would hear Trump talk with pride about Mar-a-Lago’s nondiscriminatory policy, but wondered if it was a business strategy: “Was he smart enough to realize that Palm Beach is about 40 percent Jewish and he was not going to attract the old guard anyway?”