Last week, after Gov. Kathy Hochul of New York had theatrically laid down a “Rambo”-inspired challenge to President Trump, she found herself in Washington, face to face with the president’s deputy chief of staff, Stephen Miller.

They exchanged pleasantries that Friday morning. Ms. Hochul mentioned her desire to reschedule a canceled meeting with Mr. Trump. And that was that until several hours later, when the governor received word that the president would see her around 6 p.m.

“I had to leave my staff at the gate, my State Police at the gate,” she said in an interview on Monday. “And I looked at the Secret Service guy. I said, ‘Am I going to be OK going in here?’”

The governor’s concern seemed well placed. Just two days earlier, Ms. Hochul had lampooned Mr. Trump’s allusion to being a king and invoked “Rambo,” the Sylvester Stallone film franchise, in her vow to seek revenge for Mr. Trump’s drawing “first blood” by attempting to kill the state’s congestion pricing program.

Instead, the hourlong meeting was largely civil. They discussed at length how they both sat behind desks that Franklin D. Roosevelt, who was governor of New York before he became president, had used.

The fate of Mayor Eric Adams, who is facing federal corruption charges that Mr. Trump’s Justice Department has ordered prosecutors to drop, did not come up, she said. But they debated the state’s offshore wind program, which Mr. Trump dislikes, and she said she told him that federal immigration agents should leave immigrant families alone during enforcement operations.