Yes, they do, actually.
More seriously, I'm very glad to read this. Do the Armenian and Russian Churches enjoy especially good relations?
Not that I know of.
No lie, this is actually shocking to me. I've met many Armenians in my lifetime accross the nation & i've NEVER heard a single one of them have anything nice or good to say about Russia or the MP.
I'm willing to believe i'm wrong here, just was speaking out of my own personal experience. I should have said that in the beginning.
Anecdotes are not always indicative of reality. Nor do I really think the average layperson (or even local clergyman) is probably aware of the longstanding warm and fraternal bonds between the two churches, especially in the post-Soviet era.
The Catholicos and the Patriarch of Moscow have enjoyed several very warm and cordial visits in Russia and Armenia. When Pat. Kirill was elevated, the Catholicos was rather prominently in attendance, and the MP has sent similar delegations to major church events in Armenia. There are many Armenian parishes in Russia that serve the rather large Armenian population there, many of whom have moved to Russia over the past 10 to 15 years for largely economic reasons. There are even several MP parishes in Armenia, which predominantly cater to Russian soldiers and various ex-pats. The link between the two countries is particularly personal for the Catholicos, as he not only studied in Russia for a time, but his brother (also a bishop) is the primate of the Armenian Diocese of Russia, and resides in Moscow.
So, yes, to answer your original question, the churches do "enjoy especially good relations."