"The Only Begotten" is referring to the incarnation of Christ, Him becoming a Man, the God Man.
I can't explain how God the Holy Spirit did this! I don't know what form Christ was when this happened. I don't know how Christ was in the beginning with God the Father. But I do know that Christ has been from Everlasting with the Father, He as God the Father has no beginning!
Let me show you something that just came to mind, something I was shown and taught many years ago in Bible school. I will let you be the judge of what I show you! I'm pushing nothing on you, just want you to be aware of this scripture.
Hebrews 7:1-3 KJV
"For this Melchisedec, king of Salem, priest of the most high God, who met Abraham returning from the slaughter of the kings, and blessed him;
To whom also Abraham gave a tenth part of all; first being by interpretation King of righteousness, and after that also King of Salem, which is, King of peace;
Without father, without mother, without descent, having neither beginning of days, nor end of life; but made like unto the Son of God; abideth a priest continually."
We know that Melchisedec was a real person, Paul said in vs. 4, now consider how great this man was...
We know that this man cannot be the King of Peace, as Paul said, this must be a reference to Christ.
We know that the reference to this man "without father, without mother, without descent..." is referring to no records of his birth.
So we come to the conclusion that this man, Melchisedec, was born with no records of genealogy, no record of birth, to serve as a type of Christ. Chosen by God as a Priest of God in the order of the coming Christ.
If this is true, the proof of the Son of God having no beginning and no end is proven here in vs. 3.
"Without father, without mother, without descent, having neither beginning of days, nor end of life; but made like unto the Son of God; abideth a priest continually."
Christ was not born of procreation, the womb of Mary was just a type of shelter for Him to grow from the seed of the spirit, not man. = No father, no mother, no descent.
You decide for yourself.