Mr Hogan made the claims in a Washington Post editorial savaging Mr Trump's leadership during the coronavirus pandemic.
According to Mr Hogan, the remarks were made during a private dinner hosted by the Republican Governors Association. The governor recalls Mr Trump talking about how much he respected Chinese President Ji Xinping, how much he enjoyed playing golf with Japan's Prime Minister Shinzo Abe and how well he'd gotten along with Kim Jong-Un, the dictator of North Korea.
"Then, the jarring part: Trump said he really didn't like dealing with President Moon from South Korea. The South Koreans were 'terrible people,' he said, and he didn't know why the United States had been protecting them all these years," Mr Hogan wrote. "'They don't pay us, Trump complained.'"
Mr Hogan recalled watching his wife's reaction to the president insulting her home country.
"Yumi was sitting there as the president hurled insults at her birthplace. I could tell she was hurt and upset. I know she wanted to walk out. But she sat there politely and silently," he wrote.
The broader editorial recounts the Hogans' efforts to secure test kits for their state. The couple scrambled to find test kits in the days after Mr Trump announced the federal government would leave the responsibility of coronavirus testing in the hands of the state.
Ms Hogan pleaded with the South Korean ambassador to the US, Lee Soo-hyuck, for help securing the kits.
"That request set in motion what we called Operation Enduring Friendship, 22 days of vetting, testing and negotiating an unprecedented set of protocols. Our scientists and doctors spoke to their scientists and doctors," Mr Hogan wrote. "Eight Maryland government agencies got involved, as did their counterparts in Korea. It took dozens and dozens of phone calls, night after night -- sometimes it seemed like all night -- working through language barriers and a 13-hour time difference."
Eventually, the couple secured 500,000 tests with the help of the South Korean government. Despite securing testing on their own - per Mr Trump's instructions to the states - the president was less than congratulatory to the first couple of Maryland.
"The governor from Maryland didn't really understand [about testing]," Mr Trump said. "The governor of Maryland could've called Mike Pence, could have saved a lot of money ... I don't think he needed to go to South Korea. I think he needed to get a little knowledge."
Though Mr Hogan's piece is ultimately about he and his wife's relationship with South Korea, it doesn't pull punches to blame the Trump Administration directly for leadership failures, especially in the early stages of the pandemic.
"So many nationwide actions could have been taken in those early days but weren't. While other countries were racing ahead with well-coordinated testing regimes, the Trump administration bungled the effort," Mr Hogan wrote. "The test used by the federal Centers for Disease Control and Prevention early on was fraught with inaccuracies, and onerous regulations hindered the nation's private labs. The resulting disorganization would delay mass testing for almost two months and leave the nation largely in the dark as the epidemic spread."