The daughter of Montgomery County Executive Charles W. Gilchrist came home from school recently armed with a flier that had been passed around among her classmates: It urged public school parents to write to Gilchrist to pressure him to increase his proposed 1984 education budget.

Gilchrist recounts this incident with a tone of bemused irony. Less amusing to him, however, is the growing number of missiles being launched this year in the annual war between the county government and the school system over the budget for public education.

"It is a complete PR war," Gilchrist complained last week, citing his daughter's school notice as an example. "The unfortunate thing is that the rhetoric they school officials use implies that we are doing a real hatchet job here. Historically, they don't negotiate. They go over there to the County Council, which must approve the budget, with their big staff. There is no give and take."

Gilchrist's complaints are countered by equally harsh assessments in the school system's central office and on the school board, which for the first used an audio-visual program with a narrator to argue for its budget in the County Council chambers.

From Superintendent Edward Andrews to school board president Blair Ewing to teachers in the schools, the refrain is now familiar: The school system, they say, is still reeling from school closings and controversial integration issues and needs evidence of strong support from the county executive. That evidence, they say, should be more money for teachers who were promised a salary increase and for teacher aides in some parts of the county where classrooms are bulging with students from all over the globe.

Gilchrist's budget proposal of $365 million is still $7 million short of what school system officials say they need to pay a negotiated salary increase to teachers and to maintain classroom support personnel. They complain Gilchrist arbitrarily set a budget mark that is unrealistic.

"It's a meat-ax cut," school system spokesman Ken Muir said. "It's a mindless thing. There is no rationale for it at all. It's really getting to be ludicrous."

But Gilchrist and his aides say the budget is in line. They say they are determined to hold the schools, which traditionally have been given preferential treatment by the executive and the County Council, to the same standard as other county agencies, which have been pinched increasingly by the county's shrinking resources.

"They are getting 98 1/2 percent of what they want, and that is not a bad average," Gilchrist said. He said he is increasing the budget in special cases: education for the handicapped, English as a second language and two schools with heavy minority enrollment in the lower part of the county, Takoma Park Junior High School and Montgomery Blair High School.

Although the surface argument appears to be largely over money, political considerations also are clearly involved.

Gilchrist promised in his reelection campaign not to raise taxes. But Ewing maintains the public would support a modest tax increase if it was necessary to bolster the school system.

"There isn't any other function that is carried on with tax support that is as important to the quality of the county as the public schools," Ewing said. "I believe that this is maybe the most crucial year in the last decade for the schools ."

Gilchrist says he is sympathetic to the problems of the schools and he agrees with the board's overall philosophy. But he said there have to be limits. He is supported by much of the County Council, including his adversary, president David Scull.

Scull's gripe with the school board is over a pay raise negotiated with teachers that amounts to an 8 percent increase, bigger than for any other county employes. He said some of that money could be used for other school services, such as teacher aides.

The school board majority, however, which owes its electoral successes in part to the teachers' union, is adamant about fulfilling the obligations already negotiated with teachers. School board members say it is unlikely the majority would vote to break the contract with the teachers and reopen negotiations.

The County Council ultimately will set the budget figure for the school system. The deadline is May 15, so the public relations war may continue unabated for another month.

What it all boils down to is a new kind of battle in Montgomery County: a fight between neo-liberals (Gilchrist, Scull) and traditional liberals (Ewing, school system officials) over who gets the biggest portions of the pie and how big the pie should be. The fight probably will continue until the county returns to a period of financial growth.