What was the trigger for the first coming of Christ in the fullness of time? Was it the building of the second temple?

It had nothing to do with the temple. It had to do with God extending his mercies to Israel until they were absolutely exhausted. Look at Israel's history: even in Egypt God says they were sinning in every way, yet he kept his promise to Abraham and brought them out spectacularly, and the first thing they did in the desert was worship another god, so they wandered forty years and none who left Egypt entered the land except Joshua and Caleb and those under twenty. Then through the kings they grew so wicked that the ten northern tribes were carried off into Assyrian captivity and scattered. Judah and Benjamin did not learn the lesson and went into seventy years of Babylonian captivity, and came back a small remnant that barely knew its own faith. By the time Jesus was to be the babe in Bethlehem, there was almost nothing left in Israel that could be associated with the God of Moses; they were that corrupt. So the change was made: God took the faithful remnant of Jews, made them the Church, the Gentiles poured in, and the new foundation was built on the stones that had been thrown down. That is why Christ came when he did, because of the demise of Israel and the replacement of the old covenant with the new covenant of grace.