Sadly, the answer to this is no. There were no Soviet Trekkies during the 60s and 70s. And the reason for that is simply: Star Trek the Original Series never aired in the Soviet Union.

This had various reasons: For one, the thinly veiled Cold War allegories in the original series with the Klingons as some sort of Soviet equivalent and the Romulans as a sort of Chinese stand-in would not be well received in terms of Soviet propaganda culture. Furthermore, the Breshnew era, which began with Breshnew becoming first secretary of the Central Committee was a time that might have been marked with a certain kind of detente internationally but that generally is remembered and described as a "golden time of stagnation" internally in the Soviet Union. On the one hand, it was marked by the lack of internal turmoil in the vein of Stalinization, de-Stalinization, war in Afghanistan, Perestroika and so forth; on the other hand it was marked by a distinct wide-spread feeling of stagnation and lack of progress economically, culturally and so forth. Even overseeing the Cold War metaphors in the original Star Trek, the whole feel of the series as this final frontier, space adventure did distinctly not fit the cultural policies and the overall feeling of the period.

Also, and this was probably a much bigger problem for any Western TV show to air, Soviet TV knew no such thing as a TV show, meaning an ongoing and yearly returning series with an overarching plot. In terms of narrative entertainment, Soviet television was marked by a format that is most comparable to what today we know as a mini-series, meaning movies with several parts and returning actors.

The first such mini-series aired in 1965 and was called Вызыва́ем ого́нь на себя́ (Vyzyváem ogón' na sebyá – something like "Calling fire on ourselves", sorry I only have secondary Russian skills, in English it is commonly translated as Calling for fire, danger close). It dealt with the events in the Bryansk region during the Second World War, following the story of a woman named Anna who joins the Partisans under the moniker "Rezeda". She and her comrades take over a German air field for Soviet Operations to culminate in a heroic battle for said airfield during which many of the Partisans, who by that point in the series included also Czechs and Poles, die heroically, hence the title of the series. It aired in four parts and was what today would be described as "inspired by real events".

The most popular Soviet mini-series however that is still referenced positively today was called Семнадцать мгновений весны (Semnadtsat' mgnoveniy vesny - Seventeen Moments of Spring). It aired in twelve parts in 1972 and also dealt with Soviet WWII history (a returning theme for the period that heavily relied upon the victory in WWII, creating a sort of mythical moment to legitimize Soviet Rule) but from a perspective markedly different than that of Calling Fire or other movies at the time. The series essentially dealt with the exploits of Soviet James Bond.

Vyacheslav Tikhonov is Maxim Isaev, better known by his cover name Max Otto von Stierlitz. Hunted by the brutal SS-officer Kurt Eismann, Isaev must foil various evil Nazi schemes by the notorious Walter Schellenberg and other famous Nazis up to and including Hitler. Posing as an officer of the Amt SD-Ausland Isaev learns of a plot by the Nazis to make a separate peace with the Allies by sending Schellenberg to Switzerland and he is the only man in a position to stop it. On his way to foil the plan, not only must he outmaneuver the famous head of Gestapo, Müller, but also his nemesis Eismann. Will he succeed in getting evidence to Stalin so that he can confront Roosevelt about it?

While the series was initially part of what the West would call a public relations campaign for the Soviet State Security, the series was a smash success with an estimated 50-80 million viewers per episode. It is said, that to this day, it remains the most quoted Soviet movie production of all time.

But Soviet Television in the Andropov years also featured Soviet Science Fiction mini-series. Priklyucheniya Elektronika or The Adventures of Elektronik was a children's production that details the adventures of a robot who escaped from his lap and found the child, Sergey, on which he was modeled, meaning they look exactly alike. But more successful and with a deeper cultural impact was 1985's Gostya iz budushchevo (Guests from the Future), another children's Sci-Fi program that aired in various installments. It features, you guessed it, a girl from the future traveling all the way back to 1984 to take a friend she makes there back to her time of 2084. It featured teleporters, aliens, space flight, and anti-gravity as well as some other tropes that resembled Western ideas of Sci-Fi.

And while Giest from the Future does have a distinct Soviet vibe, this is not coincidence as the Gorbachev era of glasnost (openess) included a greater openeness of the Soviet Union for Western cultural and pop-cultural media. This not only lead to the Scorpions performing in the USSR but also – and this brings us back to the subject of Star Trek – to the first time Soviet citizens would see Star Trek.

On June 26, 1987 citizens of the USSR would lay eyes upon a Star Trek movie for the first time. As part of a cooperation between the World Wildlife Fund and the team of Star Trek, Leonard Nimoy, himself from a family of Jewish Ukrainians, traveled to the Soviet Union to show Star Trek IV: The Voyage Home to a selected Soviet audience in Moscow.

The Washington Post wrote on June 29:

Before "Star Trek IV" premiered here last weekend, Soviets thought of Chekhov as a writer of Russian short stories and Spock as a doctor for American babies.

In a country deprived of the first three "Star Trek" feature films, disentitled to two decades of reruns of the beloved American television series and thus rendered naked of so-called "trekkies," the Moscow debut ran the risk of losing a cult in translation. (...)

In showings to select audiences here, however, it turned out to be the humor that turned Muscovites into would-be "Trekkies," despite the sudden transplantation of jokes from an American to a Soviet context.

Or maybe because of it. (...)

Overall, the movie drew the kinds of praise, giggles and ovations that touched off the Trekkie cult across the American continent.

Tatyana Sherbina, a young Soviet poet, summed up the Soviet public reaction by calling the movie "silly but adorable." Sava Kylish, a Moscow film director, agreed. "It was funny," he said, "something people here can relate to."

So there you have it. No widespread movement of Trekkies existed in the USSR because they couldn't see it and their television was structured differently when it came to shows. Star Trek movies and TV shows only existed, if at all, as bootlegs from the West and those never gained enough circulation to build up something like a fandom.

Also, to address some myths that float around the internet concerning Star Trek and the Soviet Union: Pavel Chekov was not included because of a Pravda editorial because said editorial just seems to not exist at all. Also, there never was a Soviet TV show called Kosmicheskaya Militsiya or Cosmic Militia with a spaceship called Red Adventurer and a Captain called Comrade Commanderto. That, unfortunately, is a hoax and wouldn't have worked at all for above explained reasons.

Sources:

• Richard Stites: Russian Popular Culture: Entertainment and Society Since 1900.

• Anthony Olcott: Russian Pulp: the Detektiv and the Russian Way of Crime.

• Adele Marie Barker: Consuming Russia: Popular Culture, Sex, and Society since Gorbachev.

• J. M. Dillard, ‎Susan Sackett: Star Trek, where no one has gone before: A History in Pictures.