The Fairfax County Board of Supervisors, citing a sharp increase in reports of child sex abuse, unanimously approved a $412,000 program yesterday that will include hiring 18 social workers trained to investigate abuse cases.

Fairfax County Executive J. Hamilton Lambert said the action shows that a county government accused of listening only to the "shouts of the subdividers" also reponds to "the quiet whimper of the abused child."

"We're going to have to spend some money," said Supervisor Elaine McConnell (R-Springfield), "but it will be well spent if we protect our children."

The program, which Lambert called a comprehensive effort, is expected to be in effect by mid-February. It calls for a concerted attack by various county agencies, including social workers, police and school authorities and the immediate creation of a child abuse task force, to be composed of citizen group representatives, elected and appointed officials and two nationally recognized experts on child abuse.

The program will focus on public education, law enforcement and treatment of offenders and their victims. The County's Department of Social Services is to use its 18 new staffers to help investigate abuse and neglect complaints and provide more information to the public about child abuse.

The police department will add four new positions in a new child service section within the Criminal Investigation Bureau that will focus on cases of child abuse and missing or runaway children.

Since July 1982, 101 cases of physical abuse have been referred to the county police, Fairfax officials said. During the same period, 147 sex crimes involving youths were referred to the police sex crimes division, officials said. During the past two years, the department has averaged 1,347 reports of children who are missing or have run away.

Despite the board's concerns, some county officials said yesterday that the supervisors' fears may be exaggerated. "We really don't have a big problem in Fairfax County, to our knowledge," said Fairfax County Police Chief Carroll D. Buracker. "But we would just like to be in a position to deal with the problems as they surface and do a bit more thorough job."

The supervisors in September said they were prepared to provide police with extra money to investigate what they viewed as a growing number of cases of child molestation and voted 7 to 0 to step up efforts to prevent child abuse. Their action followed an attack on an 11-year-old girl assaulted while walking to school in West Springfield and arrests related to child pornography.

The board's action yesterday came as state Sen. Joseph V. Gartlan Jr. (D-Fairfax) said he would press for legislation making it easier to prosecute sex offenders. He called known instances of child abuse "the old tip of the iceberg."

Last year Gartlan proposed a similar bill that would have let prosecutors use in court the first statements children in sex-abuse cases make to police or social workers. Often, he said, children are persuaded by their abusers, parents or others to change their stories. The bill last year passed the Senate, he said, but died in a House committee.

Virginia Ratliff is coordinator of the county's Victims' Assistance Network, which this year has staged programs to increase awareness of abuse in 20 to 25 of the county's public schools. "I think the whole system is responding very responsibly to a growing problem," she said.