What is the Biblical basis that the second Person of the Trinity was the Only-begotten Son of the Father before His incarnation? Do we have any proof from the Bible that the second Person of the Trinity was the Only-begotten Son of the Father before His incarnation?
The classic Bible verse to clearly declare that Jesus was the only-begotten son of God before the incarnation is John 1:1-2, declaring Jesus to be fully God, eternally with his Father.


  1 In the beginning was the Word, and the Word was with God, and the Word was God. 2He was with God in the beginning. (John 1:1-2).


From this eternity:


  14The Word became flesh and made his dwelling among us. We have seen his glory, the glory of the One and Only, who came from the Father, full of grace and truth. (John 1:1-2).


Notice the words ‘One and Only’.

So Jesus is the Eternal Word, of the Father and is God of very God.  Now the eternal word of God is the express image ‘coming out’ of the Father which is his One and only Eternal Son.

God the Father Himself declared this Jesus to be his eternal Son, during the Baptism of John:


  9At that time Jesus came from Nazareth in Galilee and was baptized by John in the Jordan. 10As Jesus was coming up out of the water, he saw heaven being torn open and the Spirit descending on him like a dove. 11And a voice came from heaven: “You are my Son, whom I love; with you I am well pleased.” (Mark 1:9-11)


Now the only-begotten Son is not a fictitious rapture of his human nature in the incarnation, or resurrection but from eternity. Jesus said:


  And no man has ascended up to heaven, but he that came down from heaven, even the Son of man which is in heaven.” (John 3:13)


I will end it here as this subject has been fully proven by so many theologians through history that I would never be able to add anything to it. John Owen (Quoting St. Jerome) indicates to me that we need not fuss over the term only-begotten. The Messiah, the Son of God has so many names referring to the same 'One and Only' as always believed in the Chruch:


  And Jerome, speaking of the effects of this mystery: (Comment. in
  Ezekiel, cap. 46:)
  “Ne miretur lector si idem et Princeps est et Sacerdos, et Vitulus, et Aries, et Agnus; cum in Scripturis sanctis pro varietate causarum legamus eum Dominum, et Deum, et Hominem, et Prophetam, et Virgam, et Radicem, et Florem, et Principem, et Regem justum, et Justitiam, Apostolu, et Episcopu, Brachium, Servum, Angelum, Pastorem, Filium, et Unigenitum, et Promogenitum, Ostium, Viam, Sagittam, Sapientiam, et multa alia.”
  — “Let not the reader wonder if he find one and the same to be the Prince and Priest, the Bullock, Ram, and Lamb; for in the Scripture, on variety of causes, we find him called Lord, God, and Man, the Prophet, a Rod, and the Root, the Flower, Prince, Judge, and Righteous King; Righteousness, the Apostle and Bishop, the Arm and Servant of God, the Angel, the Shepherd, the Son, the Only-begotten, the First-begotten, the Door, the
  Way, the Arrow, Wisdom, and sundry other things.” (John Owen’s Works Vol 1, P39)


Yes, He was the only begotten Son of God before the incarnation. This has always been one of the core beliefs within both Catholic and Protestant churches, across all the ages.