No. The Soviets never attempted to claim that the Apollo Moon landings did not occur. However, they did try to downplay the significance of the event.

The reaction to the Moon landings was actually quite positive in the Soviet scientific community, though still certainly muted or downplayed significantly to the Soviet public at large. In the press, much more attention was definitely given to the Soviet unmanned Luna exploratory probes than to America's Apollo Program successes, but there was no attempt to question or hide the fact in the USSR however that the first manned landing on the Moon absolutely did happen, and that it was obviously a monumental achievement. The scientific minds in the USSR certainly recognized it as such. This massive global story was also duly reported by the state-run press to the Soviet citizens, and while they presented the facts to the public accurately (though sparsely), they were sure to downplay its significance to America's and the USSR's standings in the "Space Race". The Soviet press sold the story to the citizens of the USSR as a victory for humanity, not merely for the USA.

There is a decent account of the Soviet scientific, political, and press reaction to the first Moon landing in the NASA archives, found in a NASA document published at the end of 1969 that was titled "NASA's Astronautics and Aeronautics, 1969 - A Chronology on Science, Technology, & Policy". Here is a link to the document, in PDF format.

In this document, which tracked national and international news stories related to space program activities, it states the following chronological breakdown during and initially following the Apollo 11 mission (beginning on page 233 of the PDF file).

July 16: U.S.S.R. radio and TV gave factual accounts of Apollo 11 launch but maintained third day of silence on Luna XV [the Soviet unmanned probe also headed for the Moon at the same time]. Major Soviet news program at 8:30 pm Moscow time showed tape of Apollo 11 liftoff taken from live comsat coverage.

Immediately after the successful touchdown of Apollo 11 on the Moon, this same document contains the following breakdown of Soviet reaction (on page 241 of the PDF file)

July 20-21: Soviet Premier Alexsey Kosygin complimented U.S. on lunar landing and expressed interest in widening U.S.-U.S.S.R. space cooperation during July 21 Moscow discussion with former Vice President Hubert H. Humphrey, who was ending Soviet visit. Soviet TV did not carry live coverage of Apollo 11 lunar landing July 20; Tass announcement was read by newscaster and carried in two-paragraph item on Pravda's front page. Evening paper, Izvestiya, accorded story more space and featured photo of astronauts on the moon. On TV, Cosmonaut Konstantin P. Feoktistov described landing as "major landmark" and said crew has coped "brilliantly" with mission. Georgy Petrov, Director of Soviet Institute for Cosmos Research, called Apollo 11 "outstanding achievement" but said more data per ruble could have been gathered by unmanned probes."

A few days later, on July 24th, after the Apollo 11 crew had safely returned to Earth and splashed down in the Pacific Ocean, the same document (on page 252 of the PDF) reports the following -

"In U.S.S.R. TV viewers had live coverage for first time during mission at Moscow TV station hooked into Eastern Europe's Intervision network for live transmission of astronauts being deposited on carrier Hornet. Later, station devoted first two-thirds of final newscast to Apollo 11 and announced that Soviet President Nikolay V. Podgorny had sent telegram to President Nixon offering "our congratulations and best wishes to the space pilots." "Soviet Academy of Sciences president Mstislav V. Keldysh called voyage "a big contribution to space exploration and further progress of world science." Cosmonauts send message to Apollo 11 crew: "We... closely followed your flight. We wholeheartedly congratulate you on the completion of your wonderful journey to the moon and safe return to earth."

Sergei Khrushchev, who is the son of Soviet Premier Nikita Khrushchev, has been interviewed about this very subject of the Soviet feelings about the Moon landings by Scientific American, and here is a link to the Q&A article that you might find interesting, where Sergei talks specifically about Soviet reaction to the American success of Apollo 11.

So no, the Soviet Union never challenged the American claims of landing on the Moon. The Soviets of course also had their own Deep Space monitoring equipment, and while not as advanced as NASA's (along with associated allied nations like the UK and Australia) system that comprised the American-backed Deep Space Network utilized during Apollo, that Soviet gear allowed the USSR to independently follow the Apollo missions to the Moon and confirm that the astronauts had indeed landed up there and were sending signals (radio, TV, and telemetry) from both lunar orbit and the lunar surface. The Soviets knew the lunar landings were technically possible (they had a successful unmanned lunar landing program of their own at the time, as well as an ultimately unsuccessful manned effort they were pursuing as well), so after the landings, the Soviet public efforts just focused on trying to mitigate the American geo-political achievement that the Apollo Program gave the USA by downplaying its importance, but they never attempted to deny that the Apollo landings happened.