During his first visit to the Kennedy Center since making himself the chairman of its board, President Trump had a lot to say about Broadway shows, dancers in silk tights, the Potomac River and Elvis Presley.

But in a private discussion at the start of a meeting of the center’s board on Monday, Mr. Trump offered something he usually steers away from in bigger settings: a personal anecdote about his childhood.

He told the assembled board members that in his youth he had shown special abilities in music after taking aptitude tests ordered by his parents, according to three participants in the meeting.

He could pick out notes on the piano, he told the board members, some of whom he’s known for years and others who are relatively new to him. But the president said that his father, Fred Trump, was not pleased by his musical abilities, according to the participants, and that he had never developed his talent.

“I have a high aptitude for music,” he said at one point, according to people at the meeting. “Can you believe that?”

“That’s why I love music,” he added.

Mr. Trump’s remarks have not been previously reported. They were not part of an audio recording of the board meeting obtained by The New York Times earlier this week.

But they are a story he has told in private, according to a person with knowledge of the comments, about a period in his life before his parents sent him to the New York Military Academy at the age of 13.

The anecdote came as a surprise to some of the people in the room.

The country singer Lee Greenwood, known for “God Bless the U.S.A,” confirmed in an interview that Mr. Trump had made a point of highlighting his childhood connection to music at the meeting on Monday. Mr. Greenwood, who was appointed to the Kennedy Center’s board by Mr. Trump, has produced a version of the Bible with the president. He said that the two had discussed the president’s abilities before.

“He’s absolutely very creative and very artistic,” Mr. Greenwood said. “I do not doubt that he has a great ear for music.”

Asked about the anecdote, Steven Cheung, the White House communications director, did not directly address it but said that the president “is a virtuoso and his musical choices represent a brilliant palette of vibrant colors when others often paint in pale pastels.” As the president and Kennedy Center chair, Mr. Cheung said, “there is nobody more uniquely qualified to bring this country, and its rich history of the arts, back to prominence.”

Mr. Trump has long shown an interest in music and theater, and he once dreamed of becoming a Broadway producer. At the meeting on Monday, he polled board members on which was better: “The Phantom of the Opera” or “Les Misérables.” He reminisced about attending the premiere of Andrew Lloyd Webber’s “Cats” in the early 1980s. And he spoke of his love for singers like the Broadway star Betty Buckley and musicals like “Hello, Dolly!” and “Fiddler on the Roof.”

“So many great shows,” he said at one point, according to the recording. “So many great shows.”

Mr. Trump moved to oust the Kennedy Center’s previous board chairman, the financier David M. Rubenstein, and all the board members appointed by the Biden administration last month. He had told allies for weeks that he wanted to lead the Kennedy Center, which he has occasionally incorrectly referred to as Lincoln Center, the premier arts venue in his hometown, New York City.

Mr. Trump disparaged the current hit musical “Hamilton” as he toured the center on Monday; its producers canceled a planned tour of the Kennedy Center next year to protest his takeover of the institution, which had been bipartisan for decades. Other artists also canceled engagements there.

The Kennedy Center has hosted and bestowed honors on performers who have been critical of Mr. Trump’s behavior as president, a fact that he and his advisers often mention.

Mr. Trump’s desire to influence programming at one of the leading U.S. arts centers has not surprised some of his longest-serving aides.

Mr. Trump spends hours working on the playlists for his rallies, and blasting music on an iPad at his clubs, as he personally acts as a disc jockey. Some artists have repeatedly asked him to stop using their music.

During his first term, Mr. Trump’s advisers often found that music was a way to calm him when he was furious. On his airplanes — his private one known as Trump Force One, and on the presidential plane, Air Force One — Mr. Trump often blasts the music so loud that it pulses through the cabin.

During a campaign town hall event in Pennsylvania in October, after two attendees required medical attention, Mr. Trump stunned some of his aides by pausing the event and having his team play music for more than half an hour. Mr. Trump stood onstage swaying and dancing as the music played.

During a late-day meeting with aides going over his playlist in the first term, Mr. Trump had the group go through Spotify looking for songs from “Tommy,” the Who’s rock opera, for more than an hour in search of a specific song that he thought he remembered. The aides couldn’t find it.