After Representative Thomas Massie, an outspoken Republican critic of President Trump, lost his seat in a closely watched primary last week, he brushed off questions about his future political aspirations by saying he was going to run — back to his off-the-grid Kentucky farm.

But as Mr. Massie pondered his future, and recovered from the most expensive House primary in recent years, he set a heading this week for sandier horizons, flying to Costa Rica. Waiting for him when he landed was a familiar Trump antagonist from the Republican Party: former Representative Marjorie Taylor Greene of Georgia.

Ms. Greene, who maintains a house in the Central American country, arrived there earlier this month for a two-week birthday trip, her fiancé, Brian Glenn, said in an interview. In recent days, Ms. Greene invited Mr. Massie and his wife, Carolyn, to hop a last-minute flight to join them at the house, and the congressman arrived on Tuesday, Mr. Glenn said.

In an image that Mr. Glenn posted on Instagram, Mr. Massie and Ms. Greene stand shoulder to shoulder as they smile broadly at the beach, their partners at each of their sides. The group spent Wednesday fishing, Mr. Glenn said, as they celebrated Ms. Greene’s 52nd birthday.

“We had a good time out on the water,” said Mr. Glenn, adding that they had some “spicy” conversations about politics. Mr. Massie needed a “break” after his bruising campaign, Mr. Glenn said. “We’re happy to provide a break.”

On Wednesday night, Mr. Massie posted a video on social media showing him and Ms. Greene smiling and laughing while fishing.

The visit has brought together two longtime friends who have faced a backlash from Mr. Trump over their criticism of the war with Iran and the government’s handling of files related to the disgraced financier Jeffrey Epstein.

Mr. Massie, a seven-term congressman with a penchant for drinking raw milk and registering patents for inventions, saw his profile rise in recent months as he sparred with Mr. Trump, saying he could not fully abide by the Republican agenda if it included “bankrupting the country or covering up for pedophiles or starting another war.”

Mr. Trump responded by putting his full weight behind Ed Gallrein, a farmer and a Republican primary challenger to Mr. Massie. The president visited the district to call on Kentucky voters to oust Mr. Massie, describing him as the worst congressman in the history of the Republican Party.

In the end, Mr. Massie could not overcome Mr. Trump’s intervention, losing by about 10 percentage points in his deep-red Kentucky district. But the congressman cast his defeat as a success of sorts, telling The Cincinnati Enquirer that his opponents had “bought a congressional seat.”

“But in the process we’ve inspired a lot of people to run for state rep or city council or even Congress,” Mr. Massie told the newspaper, adding that he might run again himself. “And so I do think it was a victory, even though we lost.”

Earlier this week, Mr. Massie filed a statement of candidacy allowing him to raise money for a potential run for office in 2028, according to Federal Election Commission records.

The document suggested he would be seeking office in the House. But Mr. Massie wrote on social media that he had not decided what office, if any, he might run for in two years.

Ms. Greene, a conspiracy theorist who was once one of Mr. Trump’s most fervent supporters in Congress, resigned from office at the start of the year, saying she did not wish to endure a bruising primary against a Trump-backed opponent.

Like Mr. Massie, she had forcefully criticized the president over the Epstein files. Mr. Trump, in turn, branded her a “traitor.”