Is it true that Paul did not quote Jesus a single time in the entire New Testament (of which he wrote 50%)?
1 Corinthians 11: 23-26
For I received from the Lord what I also handed on to you, that the Lord Jesus, on the night he was handed over, took bread, and, after he had given thanks, broke it and said, “This is my body that is for you. Do this in remembrance of me.” In the same way also the cup, after supper, saying, “This cup is the new covenant in my blood. Do this, as often as you drink it, in remembrance of me.” For as often as you eat this bread and drink the cup, you proclaim the death of the Lord until he comes.
I don't know if you intend to place restrictions on what constitutes "quoting Jesus" (compared to the Gospels or Acts), but these instances are at least Paul attributing commands or words to him.
1 Corinthians 7:10-11 (NASB)
But to the married I give instructions, not I, but the Lord, that the wife should not leave her husband (but if she does leave, she must remain unmarried, or else be reconciled to her husband ), and that the husband should not divorce his wife.
2 Corinthians 12:9 (NASB)
And He has said to me, "My grace is sufficient for you, for power is perfected in weakness." Most gladly, therefore, I will rather boast about my weaknesses, so that the power of Christ may dwell in me.
Acts describes an instance where Paul, saying his final farewell to the Ephesians, quotes Jesus.
Acts 20:35 (NASB)
"In everything I showed you that by working hard in this manner you must help the weak and remember the words of the Lord Jesus, that He Himself said, 'It is more blessed to give than to receive.'"
Paul spent more time (in writing) quoting the Old Testament than he did quoting anyone who was his contemporary.
It might not be entirely reasonable to make an issue of this if we are only considering his writings. There are no historical narratives in the canon that are attributed to Paul, and that's where Jesus is, by far, the most quoted. You can probably find as many quotes or attributions to Jesus in Paul's writings as in any other New Testament epistle in the Canon.
Words of Jesus' Ministry
By all these things, I have shown you that by working in this way we must help the weak, and remember the words of the Lord Jesus that he himself said, 'It is more blessed to give than to receive.'
In 1 Corinthians 7:10-11 Paul refers to "the Lord" (which is what he usually called Jesus) giving certain instructions on divorce. They clearly correspond to words Jesus spoke in Matthew 19:8-9 and Mark 10:11-12:
To the married I give this command—not I, but the Lord—a wife should not divorce a husband 11 (but if she does, let her remain unmarried, or be reconciled to her husband), and a husband should not divorce his wife.
For I received from the Lord what I also passed on to you, that the Lord Jesus on the night in which he was betrayed took bread, and after he had given thanks he broke it and said, "This is my body, which is for you. Do this in remembrance of me." In the same way, he also took the cup after supper, saying, "This cup is the new covenant in my blood. Do this, every time you drink it, in remembrance of me."
For the scripture says, "Do not muzzle an ox while it is treading out the grain," [Dt 25:4] and, "The worker deserves his pay." [Lk 10:7]
Jesus' words directly to Paul
In Acts 9, Jesus spoke to Paul in a vision. Paul is recorded telling others about it in Acts 22 and 26
In 2 Corinthians 12:9, Paul writes that Jesus said to him, "My grace is enough for you, for my power is made perfect in weakness."
In 1 Timothy 5:18 Paul quotes both the Torah and Jesus at the same time.
You can see that Paul is quoting Jesus exactly, word-for-word in Greek from Luke 10:7.
Stay there, eating and drinking whatever they give you, for the worker deserves his wages. Do not move around from house to house.
It is obvious that Paul was not present with the other Apostles and so it would be impossible for him to "remember" what Jesus had said, however this does not preclude direct revelation from Jesus (which is what his other quotes are). But if the question is trying to cast aspersion on Paul's "fitness" or authority to write Scripture then Peter even lays this doubt to rest when he lumps Paul's writings in "with the other Scriptures" in 2 Peter 3:15-16.
Does Paul ever actually quote Jesus anywhere?
And when we were all fallen to the earth, I heard a voice speaking unto me, and saying in the Hebrew tongue, Saul, Saul, why persecutest thou me? it is hard for thee to kick against the pricks. 15 And I said, Who art thou, Lord? And he said, I am Jesus whom thou persecutest. 16 But rise, and stand upon thy feet: for I have appeared unto thee for this purpose, to make thee a minister and a witness both of these things which thou hast seen, and of those things in the which I will appear unto thee; 17 Delivering thee from the people, and from the Gentiles, unto whom now I send thee, 18 To open their eyes, and to turn them from darkness to light, and from the power of Satan unto God, that they may receive forgiveness of sins, and inheritance among them which are sanctified by faith that is in me.
This is a pretty straight forward question that deserves a straight forward answer.
In the books of the canonized New Testament, I can find no place where Paul ever directly quotes any sayings of Jesus that are recorded from any other source than himself.
On several occasions, Paul attributes to Jesus what he claims to have heard in his visions (see the above answers) but he never reaffirms any of the direct teachings of Jesus as recorded in any other source.