I believe phrases like “Divine Move” or “Hand of God” were used before Hikaru no Go, and are well-known as a concept other than just a reference to the series.
I think it’s a highly subjective thing and doesn’t necessary mean just a single move either, but rather an entirely brilliantly played game against a very strong opponent that challenges one to rise to greatness. I think when Sai expresses his ambition to play the “divine move” or achieve the “hand of God”, what he is essentially saying is that he is still hoping to play his ultimate masterpiece game, one where he demonstrates nothing short of an overwhelming strength that could only be compared to God. His unfinished business is to perform such an unparalleled magnum opus.
In recent times, another example where the phrase was used was during the Lee Sedol vs AlphaGo match. In particular, Gu Li, while providing the official Chinese commentary, exclaimed “the hand of God” when Lee Sedol played move 78 of game 4.
EDIT: here’s a video of Gu Li and other commentators making the exclamation 神之一手 on seeing move 78.
We discussed this particular move in another thread, starting here:
Go Pro Yeonwoo also made a video about it:
Relevant Sensei’s Library article: Kami No Itte at Sensei's Library