I don't think there's a source for when it started to be used, but it is definitely a later term. And we know this because we have much older books that describe a Sanhedrin type system, but do not use the word Sanhedrin. Sanhedrin starts being used in conjunction with the older terms around the time of the New Testament. So not only is Sanhedrin a Greek word, but it turns out it's probably also one of many Greek words we've used.
The closest Biblical references to something like a Sanhedrin are those in the books of Maccabees. Many scholars believe the books of Maccabees were part of the Hebrew Bible since we see them as part of the original canon of the Septuagint. This explains why the Catholic church still has the books of the Maccabees, as they canonized the Septuagint. Although at a later point the books of Maccabees were taken out of Biblical canon amongst Jews, we still rely on the historical information present in them. And although the word Sanhedrin/High Court isn't used in Maccabees, a word that describes something similar is used.
2 Maccabees 11:27-33 New American Bible (Revised Edition) (NABRE)
27 The king’s letter to the people was as follows: “King Antiochus
sends greetings to the Jewish senate and to the rest of the Jews. 28
If you are well, it is what we desire. We too are in good health. 29
Menelaus has told us of your wish to return home and attend to your
own affairs. 30 Therefore, those who return by the thirtieth of
Xanthicus will have our assurance of full permission 31 to observe
their dietary and other laws, just as before, and none of the Jews
shall be molested in any way for faults committed through ignorance.
32 I have also sent Menelaus to reassure you. 33 Farewell.” In the one
hundred and forty-eighth year, the fifteenth of Xanthicus.[a]
The word used there in Maccabees is Gerousia – "senate" or "council." This is the oldest term which was used toward the end of the Persian period (cf. Josephus' Antiquities 12.3.3 and II Maccabees 11:27). It is used in the New Testament in the book of Luke in Acts 5:21 along with "Sanhedrin." The author of Luke may have used the word Gerousia juxtaposed to Sanhedrin as a way of explaining the term to Greek speaking readers who did not use the term Sanhedrin the same way the Jews were using it. In much the same way my wife has to explain to me that a "kitchen roll" isn't a type of bread, but how British people say "paper towels."
So why did we choose that word? Most Scholars would say because it's not a historical biblical institution, and therefore there isn't a historical Hebrew word for it, which means that as this new court system "develops" so to does the word for it.