For the first fifty years after the genocide, it was not officially recognized as one by Soviet authorities. Discussing it in the press and in casual conversation were not outright outlawed or anything, but it was understood that one didn't broach the topic in public, and private commemoration was generally approached from the personal angle, such as honoring one's own deceased family. The official narrative was of the "Armenian phoenix risen from the ashes", which implied that, rather than become stuck on the past, Armenians should concentrate on working towards a brighter future. Rather than use the word "genocide", a common euphemism was "what happened in Turkey".
Of course, Armenians didn't just forget it that easily, and as the fiftieth anniversary approached, a renewed interest in the events developed in Armenia, alongside demands for genocide recognition that were linked to nationalist sentiment and irredentism towards parts Azerbaijan and Turkey. Discussion of the genocide was also at a low boil in academic discourse in Armenia, and it warmed up as the fiftieth anniversary approached. In the year of that fiftieth anniversary, it became clear that demands for recognition were mainstream, and the central state apparatus in Moscow approved plans for an academic conference on the topic and a monument — but the dedication was not to explicitly say anything about genocide, just the suffering of Armenians in WWI. On April 24th, 1965, the fiftieth anniversary of Red Sunday, twenty thousand Armenians gathered in Yerevan to demonstrate for recognition from the Soviet authorities and to demand what they saw as Armenian land beyond the Armenian SSR's borders.
Interestingly, the demonstrators couched their demands in socialist language. We might think that, since it was an illegal protest, you'd expect it to be openly adversarial to Soviet power, but there's a good argument to be made, and which I personally buy, that they strongly believed in Soviet rhetoric and based their demands on its principles. They held their demonstration on Lenin Square, but didn't deface it at all, and even shouted down a speaker who disparaged Lenin. They framed their suffering as analogous to that of the western USSR in the Great Patriotic War and of the third world in capitalist aggressions like the US presence in Vietnam. And they painted Azerbaijan and especially Turkey as enemies of socialism due to Turkey's NATO membership and its alliance with Germany in WWI.
So commemoration of the genocide became both a key part of Armenian national identity, and of their self-perception as ideologically healthy socialists. They were "speaking Bolshevik", as Stephen Kotkin put it, but, as Maike Lehmann reformulates his quip, they were speaking it with an Armenian accent.
(For more on how those two things were not as contradictory as you might think, I have another answer that I seem to love linking to. Also, I am drawing mainly on a single article here by Maike Lehman, "Apricot Socialism: The National Past, the Soviet Project, and the Imagining of Community in Late Soviet Armenia", and if you don't have access to it I would be happy to PM you a link.)
So what exactly came of the April 1965 demonstration? It was the center of discussion in the Central Committee of the Communist Party of Armenia for a while afterwards, for one, with a majority of them sympathizing with the demonstrators. In 1968, a genocide memorial monument was built on a hill outside Yerevan, surprisingly enough, the first genocide memorial in the USSR. Here's an image of the monument, for visual reference. Its central obelisk represented that "risen from the ashes" motif, but it explicitly referenced WWII memorials with its eternal flame, and its division in two parts symbolized the way Armenian lands were, to Armenian irredentists, divided unfairly.
It became a tradition for inhabitants of Yerevan to ascend the hill every April 24th and lay flowers on the monument in memory of the dead. Armenian party officials, from the 1970s onwards, often took part in the procession, although the KGB was on hand to make sure nobody got too incendiary. So although it never became official, April 24th became the commonly accepted Armenian Genocide Remembrance Day in the USSR, and functioned as the beginning of the Soviet "high holidays", as it were, which culminate on May 9th with Victory Day.
There is still a lot more to the story of commemoration that I am not as qualified to discuss, so I hope someone else will add on a little. And territorial issues remained unresolved until the end of the USSR, and I am not qualified to talk about that in any more depth, no thank ye.
