Hunter Biden’s daughter, Naomi, allegedly vandalized the United States Capitol in 2011 when she worked as a Senate page — which prompted her dad to make a groveling apology to the head of the page program, The Post has learned.

Naomi Biden, who was 17 and a student at Washington D.C.’s elite Sidwell Friends high school, landed a plum gig as a page for Nevada Sen. Harry Reid — then the chamber’s powerful majority leader.

Her grandfather, Joe Biden, was vice president at the time.

In June 2011, she and a fellow page were given a tour of the Capitol dome, a privilege ordinarily reserved only for members of Congress and their guests.

Naomi and her page pal allegedly defaced the walls around the historic dome.

“It was vandalism,” said one former Reid staffer. “I remember perfectly. On the tour of the rotunda when you go up to the top of the dome, they carved their names into the stone.”

“It’s up at the very top so you can only have a couple of people up there at a time. It would have had to have been on the tour. There is no way they could have gotten up there at any other time,” the insider recalled.

“They had to bring in architects and everything to buff it out.”

A second former Reid staffer said Joe Biden was informed of the misbehavior.

Hunter Biden — who spent much of 2011 leveraging his name and connections in the service of foreign business partners — attempted to stem the imbroglio in a desperate email to Elizabeth Roach, the longtime boss of the Senate page program.

“I am Naomi Biden’s Dad. I cannot express how sorry I am for Naomi’s behavior. I have been in meetings all day and just heard . . . what happened. As soon as Naomi is finished work today I will be speaking with her. I am very disappointed in their behavior, but understand that the intent was not malicious,” he wrote on June 30, 2011, in an email contained on his abandoned laptop.

Roach replied less than an hour later.

“Thank you for the email and for reinforcing to Naomi the gravity of the situation. I instructed the pages to write apology letters to the sergeant at arms and the deputy sergeant at arms. Naomi wrote very appropriate notes to each of them,” she said.

Roach replied less than an hour later.

Roach declined to comment. Terrance W. Gainer — the former sergeant at arms who received Naomi’s apology letter — recalled the vandalism.

“The fact that it happened is not shocking. These young teenage kids, the first time they were away from home did a variety of things. Our goal was to correct them and guide them as we could so they could be successful there.”

A person with direct knowledge of the situation insisted that the carvings had been “tradition” and that “countless other Senate pages” had committed similar acts of defacement.

There appears to have been little in the way of formal consequences. Just a few weeks later, in August 2011, Naomi Biden accompanied the vice president on a trip to China.

After her Senate stint, Naomi Biden went on to the University of Pennsylvania and Columbia Law School. Like her father, she developed a taste for representing foreign nations and went into international arbitration law at Arnold & Porter, a white shoe D.C. law firm.

A rep for Naomi Biden declined to comment.

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