According to Josephus, Pompey took Jerusalem in the third month (Sivan) of 63 BCE during the solemn fast. He then later states that Herod captured Jerusalem in the third month, on the same day, also on the solemn fast, in 37 BCE.
Is there any record in Jewish literature of this solemn fast in the third month of Sivan? Or should this be a different month?
The Whiston translation of Antiquities proposes that the fast Josephus referred to may have been the one on the 23rd of Sivan, which was traditionally believed to have originally been established to commemorate the day that King Jeroboam I banned the Kingdom of Israel from bringing tithes from the first fruits (Bikkurim) to the Temple in Jerusalem. This fast is first mentioned in some medieval lists of fasts (collectively known in modern scholarship as "Megillat Ta'anit Batra" = The Later Scroll of Fasting) that appear in works such as the Geonic-period book Halachot Gedolot (link) and Rabbi Yinon ben Tzemach's piyyut (link, lines 18-19) from the 12th century CE.
However, to my knowledge, we don't have traditional sources that tie this fast to Pompey's conquest or that of Herod.
Alternatively, a fast on the 27th of Sivan is also known from various medieval lists, as well as the late antiquity List of Fasts Inscription from the Tel Rehov Synagogue in Israel (discussed by Haggai Misgav here, p. 21). However, here again we have no known connection to Pompey/Herod.
The OP is asking two separate questions.
According to the article by Andrew Steinmann "When Did Herod the Great Reign?", p.8 the quote from Josephus is:
. . . this destruction befell the city of Jerusalem when Marcus Agrippa and Caninius Gallus were consuls in Rome on the one hundred eighty-fifth Olympiad, on the third month, on the solemnity of the fast,
The comment in the article explains the dating:
The consular year and Olympiad given by Josephus indicates that Herod took Jerusalem in 37 BCE. It was the Day of Atonement ("the fast") on 10 Tishri in the Jewish calendar, but the third month (September) in the Greek calendar.
In other words, Josephus is using Greek calendar, and Herod captured Jerusalem on Yom Kippur. If we were to believe that Herod captured Jerusalem exactly 27 years after it fell to Pompei, then we have to accept that Pompei also did it on Yom Kippur. However the dating of Josephus is unreliable, as the quoted article shows.
According to the wikipedia, Pompei's siege of Jerusalem lasted 3 months
After three months, Pompey's troops managed to overthrow one of the Baris towers and were able to enter the Temple precinct, both from the citadel and from the west. First over the wall was a senior officer in Pompey's army named Faustus Cornelius Sulla, the son of former dictator Sulla. Faustus was followed by two centurions, Furius and Fabius, who each led a cohort, and the Romans soon overcame the defending Jews, 12,000 of whom were slaughtered. Only a few Romans troops were killed.
and the historical site Livius.org states:
Hyrcanus' followers allowed Pompey to enter the lower town of Jerusalem, but Aristobulus' adherents, the Sadducees, still occupied the Temple. In the west, there was a bridge between the Temple and the city, but this had been destroyed; in the south and east, there were deep valleys. Therefore, Pompey decided to attack from the north. (They only worked on the siege dam on the sabbaths, because the Jews could defend themselves on that days, but were not allowed to attack.) When the siege dam was completed, towers were rolled towards the wall of the Temple. Catapults kept up a continuous pressure by hurling heavy stones; a battering ram broke the wall, and Pompey's soldiers entered the Temple terrace, where they started to kill the defenders. Many Jewish soldiers committed suicide, because they did want to see the profanation of the sanctuary (June/July 63).
If indeed Pompei captured Jerusalem on a fast day, most likely the fast in the OP's question is the Fast of the 17th of Tammuz or Fast of Sivan 23rd as suggested by Whiston.
