According to the Trinity doctrine was the Son created by the Father at some point?
Short answer: no he wasn't. That would be the heresy of Arianism.
In Trinitarian theology the Son is said to be "eternally begotten" by the Father. Admittedly this is conceptually sorta like being created by the father, with the additional caveat that there was never a time when the son did not exist (hence he is begotten "from eternity"), but metaphysically being begotten is completely different to being created because the son is not part of creation: he has always existed.
It's a hard topic to discuss though. Language starts to get very wonky and inexact when you start talking about transcendant stuff that fundamentally can't be talked about. God is ineffable after all
See the Nicene creed, which was formulated in the fourth century in order to condemn Arianism. It has the line
[I believe] in one Lord Jesus Christ, the only-begotten Son of God, begotten of the Father before all worlds (æons), Light of Light, very God of very God, begotten, not made, being of one substance with the Father;
Where it says "begotten, not made" you can read that as saying "begotten, not created".
The trintiarian view has fairly clearly been cited in TheIronKnuckle's response, but as noted, does with the "begotten" aspect present a point of semantic confusion. The same "begotten" concept is found in Proverbs 8. While not definitive - some circles see the cited "Wisdom" as a reference to the Son. The problem in so doing, is in verses 22-25, in which "I was formed" and "I was born" are used. Alternative translations, however allow that "Wisdom" was processed or utilized "at the beginning." The poetic form of Proverbs also goes on to state that this self aware Wisdom was there before all else. This sits well with John chapter one and the eternal, pre-existing Word:
1In the beginning was the Word, and the Word was with God, and the Word was God. 2 He was with God in the beginning. 3 Through him all things were made; without him nothing was made that has been made. 4 In him was life, and that life was the light of all mankind. 5 The light shines in the darkness, and the darkness has not overcome it.
This creative being, whether as Word or Wisdom, is not part of the creation, but integral to/with God. Again, "begotten not made."
The doctrine of the eternal generation is taught in Scripture. It is not of Graeco-Roman myths but a purely Jewish teaching of preceding the second temple period.
- Old Testament
The concept of the eternal generation of the Son originated from the doctrine of the eternal generation of the Logos (Philo of Alexandria, ca. 50 CE) which is an ancient doctrine within Judaism contained in the canonical book of Proverbs 8:22-25 (LXX) where it says that God created wisdom by begetting her. In hellenistic Judaism, Proverbs 8 didn't mean that God is not eternally wise.
Romans 16:27 teaches that God is wise and such an attribute is eternal being essential to him ("eternal power and invisible attributes" Romans 1:19-20).
Wisdom was created ad intra (from the being of God) as also expressed in Psalm 45:1 (My heart has emitted my most excellent Logos - LXX).
Justin Martyr (d. ca. 165) describes the origin of the logos(= the pre-human Jesus) from God using three metaphors (light from the sun, fire from fire, speaker and his speech), each of which is found in either Philo or Numenius (Gaston 2007, 53).
- New Testament
1 Corinthians 1:24 expressedly say that Jesus himself is the "Wisdom of God" (sophia theou).
Colossians 1:15-16 shows that Jesus has the titles and roles of the personified Wisdom from Proverbs 8 ("first born over all creation" and "through whom all things were created").
In the prologue of John, the ancient doctrine is evidently applied to the person of Jesus. John 1:1-18 says that the Logos is eternal as he is (i). above all things being the creator of all things, and (ii). the only Son of God (monogenes), eternally begotten from God.
The person of the Son of God was eternally begotten (John 1:1-18) just as the personified Wisdom was eternally begotten (Proverb 8). They were not presented in Scripture as creatures but as proper offspring of God, from his own substance.