Bart Ehrman in his book ‘How Jesus Became God: chapter 4 – The Beginning of Christology’ lays out 3 different times, in Christian tradition, Jesus morphed into the ‘son of God’.
1. At his resurrection
2. At his baptism
3. At his birth/conception
There are 2 more alternatives as well:
4. From pre-existence on earth – John chapter 1 (word was with God) (this actually can be understood a few different ways but Christians use it as justification for being God)
5. He always was – because being an Israelite is being a ‘son of God’
The truth is all of these are presented in the gospels (or at least interpreted that way) but they cannot all be true.
This means likely the term ‘son of God’ was used by Jesus during his teachings or was later attributed to him. I’m not sure this is an and/or proposition – both are true.
Jesus was Jewish and by virtue of that covenant relationship be entered into being a ‘son of God’. This is the easiest explanation to describe where the term originated since it was in common use in Judaism theology.
He was also later accredited by Gentile converts with being the ‘son of God’ in the more Roman sense (ie: like a Caesar but better). This was how they would have understood him not having the same covenant connection Judaism claimed of God (or even knowing about that idea in the first place).
So when did Jesus become the son of God? Right when he was circumcised and brought into the covenant with God. Anything after that, the other 4 examples above, are Greco-Roman ideas fused to Jesus to make him one of their own. Thus why we have 4 different time periods he becomes the ‘son of God’ – each in contention with another and morphing as time moves forward.