No, he used the word "born" for that purpose. You may have the following passage in mind:Did'nt Paul use the word begotten to refer to the resurrection?
And we bring you the good news that what God promised to the fathers,
this he has fulfilled to us their children by raising Jesus, as also it is written in the second Psalm,
“‘You are my Son,
today I have begotten you.’
And as for the fact that he raised him from the dead, no more to return to corruption, he has spoken in this way, “‘I will give you the holy and sure blessings of David.’ Acts 13:32-34
You may have thought that God "raising Jesus" refers to His resurrection. If that were the case, why would he write "And as for the fact that he raised him from the dead..." as if he were addressing a different issue? And Paul is addressing a different issue. God fulfiled the good news by "raising Jesus", that is by raising up Jesus for the purpose of introducing the good news of the Kingdom. This started right at the beginning of time when God begat Him. That's when God raised Him up for the purpose of bringing good news to man. It culminated in HIs being born as a human being, and being the first to proclaim the gospel of the Kingdom (right after His forerunner, John the Baptizer).
That's when He became the human Son. But He was the divine Son from the Beginning.Did'nt Jesus become the Son when he was born as a human?
Yes. He had many different designations.Did'nt John refer to him as the Word prior to the incarnation?
He desired to lead us to trust in His kindness, to esteem Him our Nourisher, Father, Teacher, Counsellor, Healer, our Wisdom, Light, Honour, Glory, Power, and Life...
Letter to Diognetus ch. 9
He is the Son of God, and since we call Him "the Son", we have understood that he proceeded before all creation from the Father by His power and will (for He is addressed in the writings of the prophets in one way or another as "Wisdom" and "the Day", and "the East" and "a Sword", and "a Stone", and "a Rod", and "Jacob", and "Israel")...
Dialogue With Trypho ch. 100