Fast-growing Fairfax County recently passed its cross-river rival, Montgomery County, in population, according to estimates by both counties.

Fairfax places its population at 600,700 as of the first of the year and slower-growing Montgomery says its population then was 596,800. It was the first time that the two counties agreed that Fairfax had a larger population.

Although each of the counties consider their figures official, they are at odds with figures developed by the Census Bureau, which will regards Fairfax County as less populous.

As of July 1, 1977, the latest federal figures available, the bureau put Montgomery's population at 572,900 and Fairfax at 537,700. The bureau's estimates recognize the District - with 695,000 people - as the most populous jurisdiction in the region and Prince George's County as the most populous suburb with 672,100 residents.

There is a running dispute between the local governments and the Census Bureau over whose figures are more accurate. The last word will probably come when the Census Bureau's 1980 figures, based on an actual survey, are published.

But apart from exactly how accurate the local figures are, they tend to confirm a trend that has been developing throughout the 1970s: The thrust of suburban growth has shifted from Maryland to Fairfax.

"It just proves Fairfax County is a better place to live," boasted Fairfax Supervisor Martha V. Pennino (D-Centreville), who represents the rapidly developing Herndon-Reston area."People want to live on this side of the river. This is a great place to live."

However, Fairfax officials are not uniformly ecstatic about the surge of growth, which is accelerating the demand for costly services. Last month, Fairfax Acting County Executive J. Hamilton Lambert helped devise a solution to Montgomery's sewer crisis that would permit the Maryland county to get more capacity at a regional treatment plant and thus attract more home building.

"We want to see some of our growth go to Montgomery," Fairfax Board Chairman John F. Herrity said at the time. The present rate of growth in Fairfax - about 25,000 people a year - "is not in the public interest," he said.