As I briefly discussed in an earlier thread (thanks to /u/WhoTookBibet for linking), there were no serious Soviet doubts to the veracity of the American moon landing--the USSR was aware months before that the Americans were on course to beat them in a manned lunar landing. Instead, Soviet authorities intentionally downplayed the scientific significance of a manned moon landing (note the "freer choice" in selecting surface material--indeed only the following year did the USSR recover any surface samples through an unmanned mission).

Of course, the Soviets had strived for the political symbolism of a manned lunar mission, and ultimately admitted as much in 1989. But from an earlier comment:

It would have been very easy for the Russians to bust the moon landing if it was a hoax, and considering the complete and utter failure their lunar program was, they would have busted the US out in a heartbeat.

While the Soviet Union (not strictly Russian, and indeed the Baikonur Cosmodrone was in the Kazakh SSR, though populated chiefly by Russians, Belarusians, and Ukrainians) was far from a manned moon landing in 1969, it is a stretch to say the lunar program was a failure. Luna 2 succeeded in landing on the Moon in 1959 and Luna 9 successfully performed a soft landing in 1966, with the latter returning photographs of the surface, both before the United States.

The loss of the Moon Race was perceived as distinct from that of the Space Race; and this was, broadly, accepted by Western observers of the 1970s and 1980s. Challenge to Apollo is an excellent source that frames the Space Race through a Soviet lens, and goes into the Soviet response to Apollo 11.

From an outside perspective, the direction of the Soviet space program seemed simple. While the Soviets may have been looking to compete with Apollo in the early 1960s, they abandoned that goal early, perhaps around 1964-65, and had then focused only on the development of an Earth-orbital space station. For almost twenty years, this would indeed be the dominant paradigm in understanding Soviet motives during the 1960s and 1970s. (713)

And Salyut 1 would be launched in 1971, two years before the first American space station.

While we wait for a more specific response to be written and sourced an answer by /u/Dicranurus in the following thread is worth a read:

• "In the contemporary era, conspiracy theories abound. One of the most famous is that America didn't land on the moon. Is this conspiracy purely a modern invention, or were there doubters in the 1960s as well? Did the Soviet government accept it? Did Soviet citizens believe it was Western propaganda?"