To my knowledge, save for some of Jesus's (who may or may not have been the Jesus of the NT) students and students of students, none of the early Christians are mentioned by name in Chazalic sources, and the same goes for the Church Fathers. A few debates between sages and anonymous Christians are recorded, though they are sometimes referred to as "minai" (heretics). For example, (Sanhedrin 37a):
"And this is like an incident involving Rav Kahana, as a certain heretic said to Rav Kahana: You say that it is permitted for a menstruating woman to seclude herself with a man, i.e., her husband. Is it possible to set fire to chips of kindling and not have them blaze and burn? How can the couple be relied upon not to engage in sexual intercourse? Rav Kahana said to him: The Torah testifies concerning us that we are “set about with lilies,” as the Jewish people do not breach even a fence made of lilies."
Other mentions include Yalkut Shimoni on Nach 500:4, Avodah Zara 4a, Yalkut Shimoni on Nach 992:31 and Shabbat 116.
But more commonly, the censor changed it to "Tzdoki" (we can easily tell that these "tzodkim" are not so for two reasons: a. The context shows us that these people were not Jewish. b. The term is usually used with regards to debates that took place long after the destruction of the Temple and the fall from prominence of the Sadduceans). For example (Shabbat 88):
"a heretic saw that Rava was immersed in studying halakha, and his fingers were beneath his leg and he was squeezing them, and his fingers were spurting blood. Rava did not notice that he was bleeding because he was engrossed in study. The heretic said to Rava: You impulsive nation, who accorded precedence to your mouths over your ears. You still bear your impulsiveness, as you act without thinking. You should listen first. Then, if you are capable of fulfilling the commands, accept them. And if not, do not accept them. He said to him: About us, who proceed wholeheartedly and with integrity, it is written: “The integrity of the upright will guide them” (Proverbs 11:3), whereas about those people who walk in deceit, it is written at the end of the same verse: “And the perverseness of the faithless will destroy them.”"
For more instances, see here.
But in one instance, the Talmud names one of the minim (Christians) who came into contact with Rabbi Abahu - "Yaakov Minai (the heretic)" (Avodah Zara 28a):
"But Rabbi Abbahu was an important person, and yet Ya’akov the heretic placed upon his leg a salve that was actually a poison..."
Here's some discussion on this person.
It is likely that at least some of these Christians were well-known figures, considering the usage of the term "ההוא מינאי", as though referring to a particular Christian well-known to everyone (Prof. Shmuel Krauss wrote similarly here on the "Notzrim"). It is entirely possible that some explicit mentions of these Christians did exist and were censored by the Church, because typically the sages managed to wipe the floor with these guys, so to speak. That might also explain why Yaakov is named, being that he managed to poison Rabbi Abahu.
However, we should also bear in mind that it was halachically forbidden for Jews to teach non-Jews (Chagigah 13a):
"And Rabbi Ami said further: The words of Torah may not be transmitted to a gentile, as it is stated: “He has not dealt so with any nation, and as for His ordinances, they have not known them”."
And it seems that for this reason, Jerome wrote the following about his Jewish teacher in his 84th epistle:
"What trouble and expense it cost me to get Baraninas1 to teach me under cover of night. For by his fear of the Jews he presented to me in his own person a second edition of Nicodemus."
Therefore, it is equally feasible that the sages themselves decided not to record their encounters with the Christians, or at least not in great detail.
Note: I did a quick search in later sources and so far have not seen any mention of such encounters by later rabbis. There were some rabbis who did bring the Church Fathers in their commentaries, such as the Abarbanel and Shadal. Consider this section of the answer as more of a WIP as I continue looking for other sources.
Note 2: The Church Fathers themselves did name some of the Jews they came in contact with - Origen with Rabbi Hillel, son of Rabbi Yehudah Ha'Nasi, Jerome with "Bar Chaninah", Justin Martyr with "Trypho" and "Mnesias", and so forth. However, most of the time, they merely referred to them as "the Jews" or "the Hebrews". It seems that for various reasons, they, too, did not see the need to record all of the names of the people they came in contact with.
1 Moritz Rahmer thought he may be identified with Rabbi Chama bar Chaninah. Why any sage would have willingly risked his life to teach a Christian is entirely unclear to me, although obviously there were positive results in the case of Jerome's attitude towards Jews and Jewish tradition.