The "Pentecost" event described in the Acts of the Apostles (chapter 2) is often referred to in liturgical tradition also as "Whitsunday," or the Feast of the Descent of the Holy Spirit" narrated there in Acts.  It ranks, after Easter, as the second festival of the church, and there are references to its celebration as early as the 4th century.  The reference to the event as the "birthday of the church" is, so far as I know, a more informal label, not an official one.