After nearly two years in office, AFL-CIO President Lane Kirkland is about to face his first big public challenge -- showing that the vaunted old labor-liberal political coalition is still a factor in American politics.

The proving ground will be the west front of the nation's Capitol. The time: high noon, Sept. 19.

There, Kirkland hopes to asemble thousands of trade unionists, civil rights activists and ordinary people of every hue and religion from "every corner of the nation" to protest the Reagan administration's cuts in social spending and its perceived attacks on labor and civil rights gains of the past 50 years.

The event is called "Solidarity Day," and Kirkland himself has defined its hoped-for significance.

"Solidarity Day will show those who have written off the labor movement that we have lost none of our driving force," Kirkland said. He said it will be a demonstration of labor's willingness to resist "with all our strength . . . the effort that is under way to bring back the days when our nation was two-class society -- with great wealth and privilege on one side . . . and insecurity, privation and neglect on the other."

The demonstration, conceived and directed principally by the AFL-CIO, is coming when much of organized labor and many of its allies are reeling from defeats on Capitol Hill, when the political definition of "coalition" has come to mean Democrats and Republicans as well as Democrats and labor, and when a conservative administration is claiming it has an electoral mandate to pursue its spending and tax-cut policies.

Kirkland and his executive council will meet in Chicago this week to finalize Solidarity Day plans and to assess support for the project. Nuts-and-bolts details, such as sending out "demonstration invitations" to civil rights and other groups, are being handled by a special AFL-CIO staff on the third floor of the federation's building here.

AFL-CIO officials publicly dismiss suggestions that they and theirallies must turn out a given number of demonstrators to make an impression. They have avoided speaking about numerical goals, except on the demonstration permit application the federation filed with the National Park Service July 9.

The permit says 25,000 people are expected. But federation officials and other labor leaders say privately that they would be disappointed if fewer than 100,000 people came.

"This is the first big thing Lane has done," said a major Washington-based labor consultant, speaking on background. "A lot of this is going to be a measure of Lane's leadership, as well as a test of the mobilization strength of organized labor," the consultant said. He said attracting a crowd "of about 100,000" would "help to disprove Reagan's claim that labor leaders are out of step with their members.

Charles (Charlie) Hughes, AFL-CIL Solidarity Day spokesman, said it "looks new like we might already have 100,000 people coming," but disagreed with those sources who see that as a make-or-break figure. "You can make a silly case, based on that reasoning," Hughes said. "Let's say that 40 percent of the nation shows up. The White House will then say: 'So what? Sixty percent of the people stayed home.We still have the majority. We still have our mandate,'" Hughes said.

Kirkland, Hughes and others in their camp insist that the significance of Solidarity Day goes beyond the fact that it is being sponsored by the AFL-C IO and that it is the largest, most public, most militant project undertaken by Kirkland since he assumed office Nov. 19, 1979. Paul Brock, chief spokesman for the NAACP, one of several mainline civil rights groups helping to organize the protest, agrees with that view.

"In no way should this be thought of as an AFL-CIO thing in which Kirkland is just pulling the strings," Brock said. "This is more than a labor demonstration.Our goal is to show that Reagan is wrong when he says he has some kind of a 'mandate' to cut vital social programs . . . . If the AFL-DIO had not suggested the idea, we would have suggested it ourselves," the NAACP spokesman said.

Washington, of course, has been the scene of "hobs and justice" demonstrations before. The largest of those, on Aug. 28, 1963, played a key role in the passage of federal civil rights legislation one year later. Already this year, the city has felt the marching and heard the chanting of railroad workers, coal miners and textile weavers who came to protest what they viewed as bad treatment from the current administration.

Still, the Reagan juggernaut has rolled on, obliterating opposition in Congress -- even in the Democrat-controlled House, where Reagan last week racked up a 235-to-195 victory for his three-year tax-cut bill, ensuring enactment of his entire economic package.

Perhaps, for that reason, there is no apparent consternation in the White House about the impending demonstration. "I don't think anybody here, officially, has really focused on it," said White House labor liaison Robert Bonitati.

"There's always some kind of a demonstration here, and people certainly hve the right to protest . . . but we've been very busy working on the economic package and getting it through Congress," he said.

"You know," Bonitati continued, "we had quite a number of union people working with us on that tax-cut bill. We had the Teamsters and several of the maritime unions. We were very impressed with how hard they worked."

However, AFL-CIO officials insist that Bonitati and others in the White House are "misreading the tea leaves," as one federation official put it.

Democrats, Republicans and some labor leaders now are "running with the pack" in support of the administration's programs, AFL-CIO Secretary-Treasurer Thomas R. Donahue said in a recent radio interview. But when "more and more people become aware . . . of the individual horrors that are being visited upon them" by the Reagan administration's programs, those political leaders "will respond differently," Donahue said."That's what Solidarity Day is all about," he said.