Overshadowed by other news in President Reagan's press conference was a foretaste of a lively political debate on the indexing of federal income taxes -- a battle that could blow apart the fragile bipartisanship that is now the prevailing mood in Washington.

Taxes are going up. Everyone concedes that, including the president. The only question is who will pay -- and when. Reagan has accepted a hefty payroll tax hike as part of the compromise to ensure the solvency of Social Security. His budget proposes further "contingency" taxes on both personal incomes and energy production in the altogether likely situation that we are still facing huge budget deficits two years from now.

Congress doesn't think much of Reagan's "contingency" tax plan, but is squirming over what taxes it should raise. Theoretically, it could decide to drop the 10 percent cut in individual tax rates scheduled for July.

But Reagan has repeatedly threatened to veto such a move, and many Democrats have doubts about eliminating that kind of stimulus from an economy struggling to recover from a severe recession. The Democratic leadership would like to try "capping" the third-year tax cut for the rich, as a part of their continuing effort to dramatize the "fairness issue" for the 1984 campaign. But House Ways and Means Chairman Dan Rostenkowski was simply being realistic when he conceded the other day that the third-year tax cut will go into effect.

That leaves indexing as the next target of the tax-raisers. The provision to increase the standard deduction and adjust the tax brackets for inflation was not part of Reagan's original tax package, but he has embraced it warmly.

Repeal of indexing, which is supposed to go into effect on Oct. 1, 1985, would have some significant benefits for the Treasury, even though the president tried the other night to minimize them. Depending on how much inflation they expect, administration and congressional sources estimate the first-year revenues from repeal of indexing at anywhere from $6 billion to $9 billion, rising to about $45 billion a year by 1988.

That ain't hay. There is a political temptation to go after indexing because the pollsters are saying voters don't know and don't care much about whether it happens. The concept is not a familiar one, the pollsters say; the jolt to the taxpayer is of uncertain dimenson, less visible, say, than a nickel-a-gallon on gasoline or $5 a month more for Social Security.

But Reagan's press conference sounded a clear warning that he can make indexing much more of a political issue than it is today in any of the polls.

"Let's not kid ourselves," he said. "Government has found inflation a very handy method for getting additional revenues without having to face the public and demand a tax increase. It is a tax. Government gets a profit from inflation. And I would like to see the indexing put in place to permanently take away from government the incentive to create inflation in order to get more money."

The notion that a generation of Treasury secretaries has been secretly plotting to fuel inflation in order to increase the tax flow is slightly implausible. But there is enough truth in Reagan's contention to show the political potential of making it an issue. I can just hear him saying, "The people who would repeal indexing are, in effect, guaranteeing you a tax increase every year for the rest of your life."

Kevin Phillips, in a recent newsletter, argued that indexing is a particularly good issue for Reagan and the Republicans, because the crowding of brackets in the mid-range of income means that indexing provides "the most benefit to families with incomes between $18,000 and $50,000 a year.... Indexing is the one fiscal issue that has never failed in a recent statewide vote because of its overwhelming middle class appeal," he wrote.

Joel Havemann in the National Journal notes that there is even an equity argument Reagan can employ: while the biggest dollar savings go to the middle-class, the biggest percentage savings from indexing go to those at the bottom of the income scale.

Against these appealing political propositions, there is the overwhelming fact that the government is now -- even by Reagan's analysis -- underfunded. That is why such key Republican senators as Howard Baker and Pete Domenici want indexing repealed. There is also the fact that indexing would lock in all the inequities of the original Reagan tax cut.But the Democrats who are pushing repeal ought to be on notice after Reagan's press conference: they are taking on a tough fight.