Question: "I have heard that Christ quoted the Septuagint. Is there a listing of these quotes?"
It is a generally recognized fact that "the writers of the New Testament used almost exclusively the Greek Septuagint" (Timothy Michael Law, When God Spoke Greek: The Septuagint and the making of the Christian Bible (Oxford, UK: Oxford University Press, 2013), p. 850. And we find this in quotations of the Old Testament from Christ as well.
For example, when Christ entered into Jerusalem before his Passion, and the chief priests and scribes were expressing their disapproval of the children crying "Hosanna to the son of David!", Christ said:
"Yea; have ye never read, "Out of the mouth of babes and sucklings thou hast perfected praise?"" (Matthew 21:16).
This is a reference to Psalm 8:2, which according to the Masoretic Hebrew text, reads:
"Out of the mouth of babes and sucklings hast thou ordained strength because of thine enemies, that thou mightest still the enemy and the avenger" (Psalm 8:2 KJV).
Which is close, but significantly different when it comes to the very reason why Christ quoted from this Psalm in the first place. However, when you look at the Septuagint text, we find the text exactly as Christ quoted it:
"Out of the mouths of babes and sucklings hast Thou perfected praise, because of Thine enemies, to destroy the enemy and avenger" (Psalm 8:2, LXX).
The Greek text of Matthew 21:16 and the Greek text of Psalm 8:2 in the Septuagint are identical:
"ἐκ στόματος νηπίων καὶ θηλαζόντων κατηρτίσω αἶνον."
You can find a list of Old Testament quotations in the New Testament, which compares the Hebrew and Septuagint readings here:
by Joel Kalvesmaki
You can find a similar list, with the differences highlighted, by R. Grant Jones, by clicking here.
Update: As to the question of whether Christ actually quoted from the Greek Septuagint, or simply from the Hebrew text-type that was behind the Septuagint, there is no way to know. Perhaps there was a little of both, but in either case, His quotations match the Septuagint in substance.