Prince George's County Board of Education Chairman Doris A. Eugene accused County Executive Lawrence J. Hogan yesterday of seeking "needless confrontation" with the board and making "malicious statements" regarding its financial responsibiltity.

Eugene was responding to a letter last week in which Hogan said it was virtually impossible for the county to fund the schools' proposed budget for next year. If the board and the County Council had shown the "little restraint and courage" necessary under the county's tax-limiting TRIM amendment, Hogan said, there would not be a problem.

The exchange of letters represents the first round in what is expected to be a bitter struggle over the school system's request for a $336.5 million budget. To achieve that level of funding, which school officials and board members regard as "bare-bones," the county will have to increase its contribution by 20 percent and Hogan has said that cannot be done without increasing taxes and severely cutting other county services.

The budget is only 9.1 percent higher than last year's, but largely because of declining state and federal contributions, the county would actually have to pay about 20 percent more than it did last year.

Hogan maintains that it is virtually impossible to fund such a budget under TRIM restrictions. TRIM (Tax Reform Initiative by Marylanders) limits the county's collection of property taxes to about $144 million--the same level as was collected in 1979.

This year's property tax collection is expected to be $133.9 million, which means there is only $10 million more in potential property tax revenue next year. To meet the schools' budget, the county would have to pay over $28 million more.

"I hope that the County Council and the school board have not deliberately created this situation to build taxpayer support for the repeal of TRIM," Hogan wrote. He called the proposed budget "particularly astounding when viewed in the context of a 4.3 percent decline in student enrollment."

Eugene replied that school system efforts to restrain its budget have been "successful in ways other governmental officials only dream about." She said that during the last several years the schools budget has increased "about half of the actual inflation rate measured by the Consumer Price Index." In the last four years, she said, the system has closed 43 schools, saving about $65 million, and eliminated 1,477 staff positions.

Eugene called Hogan's letter "just a crude exercise in political posturing at the expense of the students, parents and staff of our school system."