I believe Josephus gives historical information on both the time period of the destruction of Jerusalem by Babylonians and the destruction of second temple of Jerusalem in 70 AD.
Jewish Wars Book Six, Chapter 10, Paragraph 1 - AND thus was Jerusalem taken, in the second year of the reign of Vespasian, on the eighth day of the month Gorpeius [Elul]. It had been taken five times before, though this was the second time of its desolation; for Shishak, the king of Egypt, and after him Antiochus, and after him Pompey, and after them Sosius and Herod, took the city, but still preserved it; but before all these, the king of Babylon conquered it, and made it desolate, one thousand four hundred and sixty-eight years and six months after it was built. But he who first built it. Was a potent man among the Canaanites, and is in our own tongue called [Melchisedek], the Righteous King, for such he really was; on which account he was [there] the first priest of God, and first built a temple [there], and called the city Jerusalem, which was formerly called Salem. However, David, the king of the Jews, ejected the Canaanites, and set-tied his own people therein. It was demolished entirely by the Babylonians, four hundred and seventy-seven years and six months after him. And from king David, who was the first of the Jews who reigned therein, to this destruction under Titus, were one thousand one hundred and seventy-nine years; but from its first building, till this last destruction, were two thousand one hundred and seventy-seven years; yet hath not its great antiquity, nor its vast riches, nor the diffusion of its nation over all the habitable earth, nor the greatness of the veneration paid to it on a religious account, been sufficient to preserve it from being destroyed. And thus ended the siege of Jerusalem.
From First Building built by Melchizedek to the destruction of Second temple of Jerusalem temple in 70 AD - 2,177 years.
From King David to the destruction of Second temple of Jerusalem in 70 AD - 1,179 years.
Destruction of First temple of Jerusalem - First temple of Jerusalem was demolished 477 and a half years after David. But in Aramaic version of Jewish Wars Book Six (In Codex Ambrosianus 7a1), it says 477 and a half years "from after" David ejected Canaanites and set tied his own people therein.
There were only few differences between Aramaic Jewish Calendar and the calendar of the Greeks and Romans. In Aramaic Jewish Calendar, the year starts in October (Tishri Qdam) and the year ends in September (Elul in Aramaic and Gorpeius in Greek) - Book "Introduction to Syriac" by Wheeler Thackston Page 83.
Based on the above calculations, I reached two possibilities. But I could be wrong since I am not good at Math.
From King David to destruction of Jerusalem (70 AD) is 1179 years. So in order to get the time period of David, I subtracted 70 (70 years of AD) from 1179 years and I got 1109 BC. I also put 1108 BC, because I don't know if there was 0 AD between 1 BC and 1 AD during that time period. So I subtracted 71 (0 AD + 70 years of AD) from 1179 and I got 1108 BC.
1109/1108 BC - King David
1109/1108 BC - 477.5 years = around 631/630 BC.
First possibility is that First temple of Jerusalem was destroyed around 631/630 BC.
Second possibility is this below.
1109 BC/1108 BC - King David
In 2 Samuel 5:4 (ESV)- "David was thirty years old when he began to reign, and he reigned forty years."
King David's rule was from 1109 BC/1108 BC till 1069 BC/1068 BC. First temple of Jerusalem was demolished 477 and a half years "after" David according to Greek manuscripts.
1069/1068 BC - 477.5 = 591 BC/590 BC.
Second possibility is that First temple of Jerusalem was destroyed around 591/590 BC.