The question should rather be: Did Jesus pronounce the Tetragrammaton? As has been brought out, it is highly unlikely that he would have pronounced it as "Jehovah".
According to the Jewish Talmud, according to uncensored versions, Jesus Christ effected miracles by calling on “the Name”, an indirect acknowledgement that he did use God’s name. He himself declared: “I have made your name known to the people...” (cf. John 17:6, 26; 5:43). Early Coptic texts (Pistis Sophia) tells of Jesus praying to his father by addressing him by various names and incantations: Aeeiouo, Iao, Aoi, and others. In another passage, Jesus addresses his Father in the following names and words: Iao Iouo, Aoi, Oia, and others. This is an indirect indication that he did pronounce God’s Name. What was a popular first century pronunciation of the Divine Name? Norman Walker puts forth a few suggestions:
Aquila’s version, made round about 130 A.D., is remarkable for its Old Hebrew lettering of the Divine Name in the midst of the Greek text. Put into square character, what Aquila wrote was yhyh, Jâh-Jâh [cf. yâh of MT and Greek ’Iá of Aq, Sym, Theod, and Quinta of Origen’s Hexapla], the popular substitute for yhwh “Yahweh”, the ineffable Name, the very naming of which was regarded as blasphemy as far back as the third century BC, if the LXX at Lev. xxiv 16 represents current public opinion.... By the time the Mishna was compiled (c. 190 A.D.) the pronunciation had become practically JeJâ as the form yeyâ shows....
The recovery of a purer Ben Asher Text by KAHLE [KITTEL, Biblia Hebraica (Third Edition, 1945)] reveals that the Divine Name was earlier pointed yehyâh, that is with the vowels of JeJâ and not those of ‘Adhonâi. It seems to me that this vocalization supports the implication of Aquila and the Mishnaic form, namely, that in the first two centuries A.D. at least, if not later, the Divine Name was uttered JehJâh or briefly JeJâ. [i]
[i] N . Walker , “The Writing of the Divine Name in Aquila and the Ben Asher Text”, Vetus Testamentum , vol . III, No. 1, January, 1953, pp. 103, 104 .