The Soviet Union did not question that Americans had landed on the Moon in 1969; in fact, it was openly discussed on television and published in newspapers. Notably, the footage of the landing itself wasn't public, and restricted to a small number of Soviet leaders. And the newspaper coverage isn't the kind you see in the United States, with large proclamations on the front page - it was much more understated. But it wasn't hidden nor contested.

The Soviet Union wasn't particularly forthcoming about their failures, including the death of would-be cosmonaut Valentin Bondarenko in 1960. So when they failed to reach the moon, the leaders of the program pivoted to other endeavors (both real and imagined), and publicly undermine the scientific and political value of landing on the moon.

The Soviet 'conspiracy' to be found is that they weren't particularly interested in landing on the moon. It wasn't until 1989 that this was publically conceded; the Soviet Union, in 1969, was quite far off from a successful moon landing, and in the months leading up to July 20 had several technical disasters. The Soviet Union began preparing for the American moon landing in the spring, and following the successful landing was in disarray, debating how to move forward. Publically, to limit embarrassment, the Soviet Union proclaimed to only be interested in an orbital space station.

Chs 15 and 16 of Asif Siddiqi's Challenge to Apollo are a great source on Soviet responses to the landing. I'll excerpt a piece of it here that directly responds to your question:

[After the moon landing,] numerous pronouncements of the last eight months disappeared, replaced by two clear and consistent themes: the Soviet objective to the explore the Moon by automated means and the longstanding goal of establishing piloted orbital space stations in Earth orbit. Implicit, of course, in both these themes was the claim that the Soviet Union had never planned to send humans to the Moon because its program had always been geared more toward scientifically productive rather than politically motivated objectives. Academician [Anatoly] Blagonravov claimed on Moscow Radio on July 2 I that the only advantage of sending cosmonauts to the Moon was to provide freer choice in picking up Moon rocks... (697)

See also "Russians Finally Admit They Lost Race to Moon" NYT (Dec 18, 1989) for the public acknowledgement of Soviet efforts in 1968/69.

As /u/Dicranurus/ discussed the Soviets, let's get into the US side:

Some conspiracy theories have a definite origin, a "Patient Zero" if you will; for example, the Phantom Time Hypothesis wasn't around until Heribert Illig came up with it

Other conspiracy theories are more organic; with the moon landings, some Americans were skeptical immediately.

A New York Times article from December 1969 noted people who claim the landings were "staged by Hollywood on a Nevada desert”. An opinion poll from 1970 put the number who "doubted the moon landings had taken place" at less than five percent, relatively normal for a conspiracy theory. A 1970 pamphlet entitled "Did man land on the Moon?" is the first known printed source specifically aimed at moon-landing skepticism.

From Apollo 11: On the Moon: Look Magazine 1969 Special:

The almost perfectly executed odyssey of Apollo 11 seemed unreal to some. In McGehee, Ark., 81-year-old Mrs. Barbara Marion Hopkins Day was so unimpressed by the moon landing that she believed it to be a hoax, contrived for mere publicity. She slept through the moon walk on Sunday and did not turn on her television set during the entire lunar voyage. "I don't believe it," she said. "I don't believe they've ever been there."

Sources:

Atlanta Constitution (15 June 1970). Many Doubt Man‘s Landing on Moon.

Chaikin, A. (2019). A Man on the Moon: The Voyages of the Apollo Astronauts. United Kingdom: Penguin Books Limited.

Cranny, J. Did Man Land on the Moon? Johnson City, Texas, 1970.

Wilford, John Noble (18 December 1969). A Moon Landing? What Moon Landing? The New York Times.