Russia denied the entire episode took place.
Qatar's Prime Minister and Foreign Minister Sheikh Hamad Bin Jassim Bin Jabr apparently spoke to Russia's UN Envoy Vitaly Churkin prior to the UN vote on Syria saying: "I warn you against using the veto. You will lose all Arab countries"
Russia Envoy Churkin replied: "If you speak to me in this manner again there won't be something called Qatar from today. You are a guest on the (UN) Security Council, respect yourself, go back to your size, I only speak in the name of great Russia with big (countries, could also mean adults)"
The recording was apparently broadcast on France 2. Here is an Arabic transcript and here is one in French. There is no mention of this incident in the English language press.
It's interesting to note that Russia downgraded its diplomatic ties with Qatar last December following an incident on November 29 in which its ambassador to the Arab Gulf State "was attacked by security and customs officers at the Doha airport." Russia then warned Qatar of "consequences" following the incident, demanding an apology and recalled its ambassador from Doha. Russia Today, Moscow's version of Al Jazeera then started broadcasting a string of reports critical of Qatar saying in one report that it is "punching above its weight".
Here is a cartoon by RT's Vladimir Kremlev: