Am I correct in saying that the identities of the zugot are only revealed to us within masechet Pirkei Avot?
Or are the names of the zugot also disclosed to us in other masechtot of the Mishnah? If so, which masechtot of the Mishnah?
What do you mean? Of course the zugot are identified elsewhere. For example, Shemaya and Avtalyon are mentioned together in Eduyot 5:6, Yoma 35b, Gittin 57b, Pesachim 66a, and other places.
All five zugot are mentioned together in Chagigah 16a:
Mishnah: [Yossi] ben Yoezer says not to lean [on the chagigah offering]; Yosef ben Yochanan says to lean. Yehoshua ben Perachia says not to lean; Nitai the Arbelite says to lean. Yehudah ben Tabbai says not to lean; Shimon ben Shetach says to lean. Shemai'ah says to lean; Avtaliyon says not to lean. Hillel and Menachem did not argue. Menachem exited, Shammai entered. Shammai says not to lean, Hillel says to lean.
Addendum in Chagigah 16b:
הָרִאשׁוֹנִים הָיוּ נְשִׂיאִים וּשְׁנִיִּים לָהֶם אֲבוֹת בֵּית דִּין: -- The first ones [mentioned of the pair] were Princes (nasiim) and their seconds were heads of the court (avot bet din).