We don't have much sources from the era. It was, after all, the era of the Savoraim, who were mostly focused on editing and redacting the Talmud.1 The following are the earliest sources I was able to find. If I find more, I'll add them.
- There's a disagreement about the lifetime of Rabbi Elazar Hakalir. Some say he lived in the Talmudic era while others say he lived later. Ezra Fleischer was of the opinion that the Kalir mentioned the Arab Conquest in one of his piyutim, which means that he lived during that time (see here, pp. 32-33):
"תשיב תגמלם גמוליהם
תיעבו גמול לועטי גמליהם
השבעתי תדריכם בטבת כולהם
תאלת תרשיש תתן להם"
Fleischer understood the stanza to mean:
"Give unto them their earnings
[as] their camel riders hated [the mishmar of] Gamul
I have sworn [You, Hashem,] to crush them in [the 10th of] Tevet
and place upon them [all of the] crises that befell the Mikdash."
Per this interpretation (and naturally, there are those that disagree), the Kalir, at least, saw the rise of Islam as a negative thing, apparently because of the Arab cruelty to Jews.2
- As @DoubleAA rightly noted, Targum Pseudo-Yonatan (whatever time and place its author may be identified with) contains some material from Pirkei D'Rabbi Eliezer, including material that may be related to the Arabic Conquest. Just one example (30:12):
"Rabbi Ishmael said: In the future the children of Ishmael will do fifteen things in the land (of Israel) in the latter days, and they are: They will measure the land with ropes; they will change a cemetery into a resting-place for sheep (and) a dunghill; they will measure with them and from them upon the tops of the mountains; falsehood will multiply and truth will be hidden; the statutes will be removed far from Israel; sins will be multiplied in Israel...he will hew down the rock of the kingdom, and they will rebuild the desolated cities and sweep the ways; and they will plant gardens and parks, and fence in the broken walls of the Temple; and they will build a building in the Holy Place; and two brothers will arise over them, princes at the end..."
It is thought by some scholars that the emphasized portions refer to Muslim construction on the Temple Mount. This implies a negative view of the Arab Conquest (although Avigdor Shinan (תרגום ואגדה בו, p. 164, note 344) thought that the Targum was more antagonistic towards Islam/the Arabs than PDE). For some more information, see here, for example.
- On the other hand, the 9th century scholar Pirqoi ben Baboi seems to have viewed the Arab conquest positively (see here, p. 398):
"...ובאו ישמעאלים והניחום לעסוק בתורה ולקרות קירית שמע ולהתפלל..."
Translation: "...and the Yishmaelim came and let them study Torah and say the Shema and pray..."
This is in reference to the gezerot shmad during the Byzantine rule of Eretz Yisrael and the subsequent Muslim-Arab Conquest.
The Arab rule over Babylon is mentioned in Iggeret Rav Shrirah Gaon from the 10th century. Rav Shrirah states that the situation of the Rashei Galuta needing to buy their status from the Arabs caused the sages great pain.
A reference to Malchut Yishmael is made in a piyut by Rabbi Yosef Ibn Avitur who lived in the 10th century (see here, pg. 2):
"לבדו ישראל מלך מחרבן ועד מלכות ישמעאל
ומאז משלונו חתן וחותן ונתחברו בקרית אריאל
סוד פלאיך מתי תגלה אל רם על כל אל
מי יתן מציון ישועת ישראל"
Translation:
"Over Yisrael ruled from the destruction and until the Kingdom of Yishmael
And since then we are ruled by groom and father-in-law and they joined together in the City of Ariel [Yerushalayim]
The secret of your mysteries when will You reveal, mighty God above all gods
Who will bring from Tzion the salvation of Yisrael?"
In short, judging from these few sources, it seems that though Jews were not indifferent to the Arab rule (some regarding it negatively while others positively), they were pretty much indifferent to the Islamic aspect of it.
1 You can see here a compilation of some of the earliest direct Babylonian sources regarding the Muslims. They date to the 11th century and onwards. Add to that Iggeret Rav Shrirah Gaon, from the 10th century.
2 Worth mentioning that some of the Jewish Arabian tribes destroyed by Mohammed were priestly tribes, much like Gamul was a mishmar kehunah.