The government's free cheese program came yesterday to Montgomery County, one of the nation's wealthiest suburbs, but fewer than expected of the county's residents eligible for the cheese arrived to accept it.

There were few lines and little waiting at the 16 distribution centers scattered about the county. And social service workers who had braced for a repeat of the scenes that occurred last month in the District, where thousands waited in the cold for hours and many were turned away empty-handed, said they were surprised that yesterday's turnout was so light.

"We kind of think people are on strike or something," said Mary Bladen of the Montgomery County Community Action Agency responsible for distributing the 5-pound boxes of American cheese. "The response is not what we thought it would be."

Officials in charge of the cheese distribution program said preliminary tallies show that only about half of the 5,736 persons eligible for the free cheese showed up yesterday. "Montgomery County certainly has its share of low-income people," Bladen said. "I guess because we have the reputation of being a rich county, people do not think that we do."

At Park Street Elementary School in Rockville, for example, volunteers distributing cheese often outnumbered the recipients, many of whom were single mothers and elderly widows who live on a few hundred dollars a month in a county where the median annual income for a family of four was $29,600 in 1980.

"I'm really surprised because I was afraid it would be really crowded," said Julie Rose, 30, who sat clutching her box of cheese to her chest as her four children, all under age 10, alternately ran up and down the halls of the school and poked at the cheese box while waiting for a ride home.

"The kids like grilled cheese sandwiches and cheese sauce," said Rose, who is separated from her husband and receives $644.50 a month in welfare and food stamps. The $7,728 a year supports her family of five.

Rose, who pays $400 a month in rent, figures she'll use the money she saved on the free cheese--about $11--to buy more bread and milk. "I guess I'll freeze some of it," she said.

Some recipients chatted casually while waiting for a bus that would take them to their homes without charge. Others sat in the hallway of the school, staring straight ahead in embarrassed silence or intently studying a sheet of recipes, featuring dishes like "quick cheese soup" and "rice-cheese puff," which had come with their box.

"Oh please," said one white-haired widow. "I never thought I would be here and I just don't want to say anything. I'm just glad to have this."

The same cannot be said of the two young sons of Nguyen T. Phuong. Phuong, 30, and the two boys escaped from Vietnam aboard a small boat three months ago.

"They don't like American food like this, but I force them to eat it," said Phuong, who left her imprisoned husband and parents behind in Saigon. "They miss Vietnam terribly and they like Vietnamese food, but I tell them they have to eat American food because I cannot prepare Vietnamese food here."

Phuong, who was a lawyer in Saigon, is enrolled in a computer science course at Montgomery College. She said she shares an apartment with her brother and pays $200 in rent out of the $449 a month she receives in welfare and food stamps.

Montgomery County was one of the last jurisdictions in the Washington area to distribute the free cheese--surplus from a longstanding federal government program to stockpile dairy products under the government's milk price support program. Besides the District, the cheese has already been distributed in Prince George's and Arlington counties and Alexandria. Distribution in Fairfax County is scheduled for Friday.