No, Armenia was not the first country to convert to Christianity. The Kingdom of Edessa was the first country although this kingdom no longer exists. Besides the ruling elites of Armenia and Georgia were always of Iranian origin making them Armenian-not-so-much: Arsacid dynasty of Armenia - WikipediaI don't know too much about the topic of religious conversion in the Near East and Caucasus, but wasn't Armenia the first country in history to officially designate Christianity as their state religion? With that type of history, it's hard to imagine how religious conversion could ever achieve large-scale success. It would be like expecting Arabia to convert to Christianity.
Even the saint that made Armenia a Christian country was not an Armenian Gregory the Illuminator - Wikipedia