Is that correct or not? Did you say the Trinity has not always existed?
God always existed. In as much as the incarnation goes, that is what the Trinity is all about. In fact no one would have even thought of the concept "trinity" except for the fact of the incarnation.
Note the Jews never distinguished between the Father and Holy Spirit. The Holy Spirit was just the spirit of God. They were one and were never distinguished as 2 persons.
The incarnation occurred in time. Although time elsewhere doesn't correspond to time in this world, time is a measure of the persistence of existence. It makes no sense to me, that God decided infinitely long ago, to create man and incarnate. So even though God never changed, He still does things in time. He didn't decide infinitely long ago, to do an infinite number of things. That makes no sense at all.
So yes the Trinity did have a beginning as per above. That does not change God however. It simply means He had done something new. It does not mean He changed, anymore than I would if I started selling fasteners. It's simply a new experience; I wouldn't be a new spunkets.