A listing of prayers by Jesus in the New Testament can be found at this site:
http://www.praylikethis.org.uk/theprayersofjesus.html
(I can't say that the list is exhaustive, but I think it is at least representative.)
I've compared the verses listed with the New World Translation and the name "Jehovah" does not appear in any of these verses.
So, there is not one verse in the New Testament (Christian Greek Scriptures) that has Jesus uttering the name "Jehovah" in a prayer. If Jesus had meant to emphasize that his followers should pray using "Jehovah" in their prayers, then why didn't he do so?
True, Jesus does pray about God's "name," but when doing so does not refer to a pronunciation like "Yahweh" or "Jehovah." Since Jesus did not utter "Jehovah" when referring to God's "name," doesn't it make more sense that Jesus is referring to God's character, authority or reputation? We often use the word "name" in such a sense. For example, we'll speak of someone "making a name for themselves." We've also heard the expression: "Open in the name of the law!" So, when Jesus prayed in John chapter 17 saying, "I have made your name manifest" he was basically saying "I have made your character manifest," not some pronunciation.
Interestingly, the Watchtower referred to this metaphorical use of "name" in the May 1, 1973 Watchtower, page 259:
WHAT GOD’S NAME MEANT TO JESUS
God’s “name” is also important to Jesus Christ. Just before his death he prayed: “I have made your name manifest to the men you gave me out of the world . . . watch over them on account of your own name which you have given me . . . And I have made your name known to them and will make it known.”—John 17:6, 11, 26.
We are not to believe that when Jesus said, “I have made your name known” or “manifest,” he referred to only the pronunciation of the divine name. His listeners were Jews who, reportedly with the exception of the high priest, did not know the pronunciation of the Tetragrammaton, the four Hebrew letters making up the name. Then, how did Jesus, by more than pronouncing the name correctly, ‘make God’s name known’ to the apostles? Note the answer given by one noted Bible commentator:
“The word name [in John 17] includes the attributes, or character of God. Jesus had made known his character, his law, his will, his plan of mercy. Or in other words, he had revealed God to them. The word name is often used to designate the person.”—Notes, Explanatory and Practical, on the Gospels by Albert Barnes (1846).
So, as Jesus ‘explained the Father’ by his own entire perfect life course on earth, he was really ‘making God’s name known.’ He demonstrated that he spoke with God’s full backing and authority. Jesus could therefore say: “He that has seen me has seen the Father also.” God’s “name” thus took on greater meaning to his early followers. Accordingly, an appreciation of it and the Personality that it stood for should be reflected in every aspect of the Christian’s life.—John 14:9; 1:18; 5:19, 30; Matt. 11:27.
So, again, is there a single verse in the New Testament where Jesus prays using the name "Jehovah" in his prayer?