With New York City struggling to cope with a continuing influx of migrants seeking asylum, Mayor Eric Adams on Friday announced a new strategy: The city would begin paying for shelter at two suburban locations outside the five boroughs.

It took less than a day for the plan to hit a wall. Officials in one of the two counties strongly rejected the city’s relocation efforts and declared a state of emergency to thwart the attempt.

The escalation marks one of the first challenges to Mayor Adams’s so-called decompression strategy to ease the burden of housing the city’s growing number of asylum seekers, at a time when thousands more are soon expected to seek refuge in New York as a pandemic-era border policy is set to expire.

The city intended to house about 300 men in two hotels in Rockland and Orange Counties, north of New York City, providing up to four months of shelter, food and medical services.

Although Mr. Adams said that he had consulted with the state and with local officials, at least two said they were caught off guard and vowed to fight the plan.

“It felt like they were trying to do a Friday night drop,” said Teresa Kenny, the town supervisor of Orangetown, who said she learned about the plan only hours before Mayor Adams announced the move. “I feel like the mayor called me to check a box so he couldn’t be criticized for not talking to us.”

The Rockland County executive, Ed Day, said he was stunned by Mr. Adams’s plan, and moved quickly to find a way to stop it. He issued the state of emergency order on Saturday, declaring that no municipality could transport or house migrants in Rockland without his permission.

“Whatever we need to do to stop this, we will do,” Mr. Day, a Republican, said in a phone interview on Sunday, adding that the county is prepared to issue fines of up to $2,000, per violation, per day, to any hotel that accepts asylum seekers from the program. “They’re basically dumping them into a county where we’re not prepared for them,” he said.

It is unclear whether the declaration of a state of emergency will prevent New York City officials from arranging the transfer of migrants. New York City issued its own state of emergency in October, in a bid to marshal federal resources to help with the growing migrant population.

Rockland is the first county in the state, outside of New York City, to issue such an order to address the migrant crisis, said Stephen Acquario, the executive director of the New York State Association of Counties.

“We need leadership here, because this is an area of unfamiliar, uncharted waters,” he said, noting that the effort to prevent the transfer of migrants to Rockland is likely to have to be decided in court. The county also has a law that prohibits hotel guests from staying for more than 30 days at a time, which could present another challenge.

Mr. Acquario, who has spoken with representative from several of the state’s counties outside the city since the plan was announced, said that he is sympathetic to New York City’s position, but that the problem requires greater cooperation between levels of government.

“This is a homelessness problem, and it’s largely the fault of the federal government,” he said. “And here we have the lowest unit of government cleaning up the mess of the United States.”

The shift in New York City’s strategy comes after Mr. Adams said in a statement on Friday that a “vacuum of leadership” at the federal level has necessitated a new “decompression strategy,” to share the burden of housing a growing number of asylum seekers across municipalities. Over 60,800 migrants have arrived in New York City since last spring, according to city officials.

Those sent to Rockland and Orange Counties would be volunteers taken from a pool of single-adult men already in the city’s care, a city spokesman said.

Calls to the hotel in Rockland County, the 170-room Armoni Inn and Suites in Orangetown, were not answered. Piles of mattresses were stacked outside of the hotel on Friday afternoon, Ms. Kenny, the Orangetown town supervisor said, even though a wedding reception was being held at the building over the weekend.

An official in Orange County identified the other hotel as the Crossroads Hotel in Newburgh; a receptionist there declined to comment.

“This came out of left field,” said Steve Neuhaus, the Orange County executive, adding that Mayor Adams told him the Newburgh hotel would receive about 60 migrants.

Mr. Neuhaus, a Republican, also questioned how the county would handle the newcomers at the Newburgh hotel, which is on a commercial strip next to an ice-skating rink, a gym and not much else.

“We are maxed out with the homeless in the county,” he said, adding that he is also considering issuing a state of emergency to block the move.