Arlington School Board member Michael E. Brunner said yesterday that he will seek the Republican Party's endorsement of his independent candidacy for the County Board, amid increased speculation that at least one of the two GOP incumbents won't seek reelection this fall.

The terms of Republicans Walter L. Frankland Jr. and Dorothy T. Grotos, who have served on the County Board for nearly eight years, expire this year. Both declined yesterday to say whether they will seek reelection to a third four-year term on the five-member board, now controlled by Democrats.

Arlington Republicans have been suggesting that Grotos will instead run for the fulltime job of county treasurer, now held by Bennie L. Fletcher, a Republican. Fletcher, 59, who has been treasurer for 12 years, has called a press conference for today at which he is expected to announce his retirement.

Brunner fueled that expectation yesterday. "It's not my intention to challenge the incumbents," he said. ". . . I don't foresee any circumstances where the three of us will be competing for two seats."

Brunner, 40, a Republican school board appointee whose term expires June 30, also said he does not expect the County Board's Democratic majority to reappoint him to the school board. As a member of that board, Brunner has attracted bipartisan support and a number of Democrats reportedly have written the County Board urging his reappointment.

"There ought to be (a political) balance on the County Board. One party or one faction should not control all the seats," said Brunner, an associate administrator of the Farmers Home Administration and a certified public accountant

"Somebody with budget-management and financial expertise is something we can always use on the County Board," he said ". . . The inclination of the present County Board majority is to go for increased taxes. My inclination is to look at ways to economize first."

Retiring Arlington Republican Committee Chairman Jade West and 10th District Republican Committee Chairman Robert Bocek praised Brunner yesterday and said they expect others may also seek the party's endorsement if the either incumbent bows out. Arlington Republicans will select their candidates May 17.