According to the Catholic Church, when did the Church begin? I would love to have an official document that says that the Church began on such and such a date but more importantly, scripturally speaking, what scriptural event describes the birth?
The Catholic Church believes the Church (catholic Church) began at Pentecost.  Here are references from its Catechism.

2617 ... at Pentecost, for the formation of the Church, his Body.
1076 The Church was made manifest to the world on the day of Pentecost by the outpouring of the Holy Spirit. The gift of the Spirit ushers in a new era in the &quot;dispensation of the mystery&quot; the age of the Church, during which Christ manifests, makes present, and communicates his work of salvation through the liturgy of his Church, &quot;until he comes.&quot; In this age of the Church Christ now lives and acts in and with his Church, in a new way appropriate to this new age
731 On the day of Pentecost when the seven weeks of Easter had come to an end, Christ's Passover is fulfilled in the outpouring of the Holy Spirit, manifested, given, and communicated as a divine person: of his fullness, Christ, the Lord, pours out the Spirit in abundance.122
732 On that day [Pentecost], the Holy Trinity is fully revealed. Since that day, the Kingdom announced by Christ has been open to those who believe in him: in the humility of the flesh and in faith, they already share in the communion of the Holy Trinity. By his coming, which never ceases, the Holy Spirit causes the world to enter into the &quot;last days,&quot; the time of the Church, the Kingdom already inherited though not yet consummated.
830 The word &quot;catholic&quot; means &quot;universal,&quot; in the sense of &quot;according to the totality&quot; or &quot;in keeping with the whole.&quot; ...  The Church was, in this fundamental sense, catholic on the day of Pentecost and will always be so until the day of the Parousia.

To add scripture.

Acts 2:1-4, 33 And when the day of Pentecost was fully come, they were all with one accord in one place. And suddenly there came a sound from heaven as of a rushing mighty wind, and it filled all the house where they were sitting.And there appeared unto them cloven tongues like as of fire, and it sat upon each of them. And they were all filled with the Holy Ghost, and began to speak with other tongues, as the Spirit gave them utterance. ... Therefore being by the right hand of God exalted, and having received of the Father the promise of the Holy Ghost, he hath shed forth this, which ye now see and hear.