The Qur'an is intertextual in very interesting ways. To my knowledge, it's not clear to anyone how much the initial audience would have understood, but it certainly assumes knowledge. To give one very basic example, pretty much every Muslim in the world could tell you that Adem/Adam's wife is name Eve, but that fact is not expliciitly in the Qu'ran, where his wife is unnamed (it is mentioned In Hadith, however). Were these sorts of things taught alongside the Qur'an or did the early Arab already know these things? I'm not aware of any compelling evidence either way, but it seems unlikely that polytheists in Mecca or Medina would be expect to know the back stories for prophets and messengers who are mentioned in the Qur'an but whose significance is only explained elsewhere (some prophets, like Isa/Jesus and Yusuf/Joseph are explain in loving detailed).

Some of the details are clearly related to the details in the Hebrew Bible/New Testament, even when they mean little in the context of the Qur'an. For instance, the story of Yunus/Jonah I think is only related in 37:139-149, which relates all the very basics of the story... but it also includes line 37:146 which in context reads,

But we threw him onto the open shore [from the whale's belly] while he was ill.

And we caused to grow over him a gourd vine

And we sent him to [his people of] a hundred thousand or more.

Gourd vine? I have no idea how this is treated in tasfir and hadith, but it seems a clear reference to Jonah 4:6-10, after Jonah is angry at God, because Nineveh repented and so there wasn't the destruction that Jonah had prophecized. After getting into a little tiff and wishing himself dead, he's just hanging out outside the city.

6 The Lord God appointed a bush [possibly a castor bean plant], and made it come up over Jonah, to give shade over his head, to save him from his discomfort; so Jonah was very happy about the bush. 7 But when dawn came up the next day, God appointed a worm that attacked the bush, so that it withered. 8 When the sun rose, God prepared a sultry east wind, and the sun beat down on the head of Jonah so that he was faint and asked that he might die. He said, “It is better for me to die than to live.”

9 But God said to Jonah, “Is it right for you to be angry about the bush?” And he said, “Yes, angry enough to die.” 10 Then the Lord said, “You are concerned about the bush, for which you did not labor and which you did not grow; it came into being in a night and perished in a night. 11 And should I not be concerned about Nineveh, that great city, in which there are more than a hundred and twenty thousand persons who do not know their right hand from their left, and also many animals?”

So the there's a point to the bush in the Hebrew Bible, it's a lesson to Jonah that him saving those Ninevahans from punishment was good, and why God wanted to send him to Ninevah to make them repent, but it just sort of hangs there in the Qur'an without as clear a purpose (don't get me wrong, there are plenty of lines like that in Hebrew Bible as well). Further, the incident also comes after Jonah's preaching in the Hebrew Bible, but before it in the Qur'an.

So if we take for granted that the Qur'an's stories are adapted from Jewish and Christian scripture (and not, you know, a separate retelling through divine revelation, as Muslims do), then we tend to see a lot of these details that seem half remembered like the one above. But we also get references to stories where there's no details at all mentioned, but it's assumed that the audience knows some details that makes them work mentioning.

Take Dhul-Kifl, for instance, who's sometimes connected with Ezekiel (based on information in Hadith and other sources, as we shall see there's really nothing in the Qur'an). He's mentioned twice, both in lists with no details:

And (remember) Isma'il, Idris, and Zul-kifl, all (men) of constancy and patience./We admitted them to Our mercy: for they were of the righteous ones. [21:85-86]

And commemorate Isma'il, Elisha and Zul-Kifl: Each of them was of the Company of the Good. [38:48]

What do you make of that?

Somethings are clearly adapted from Christian teachings: all the references to Jesus and Mary, obviously. But they also include the story of Zechariah (John the Baptist's father) from the Gospel of Luke chapters 1 and 3, told in Quran 19:2-15, including details like being made a mute before John's birth. You know, detailed stories that seem unlikely to have been just general knowledge for pagans of the era.

There are also some things that are clearly related to Jewish teachings that weren't incorporated into Christianity. At the end of the Cain and Abel story (Quran 5:27-32; their names aren't explicitly given I believe they're just identified as "Adam's two sons") with a moral lesson:

Because of that, We decreed upon the Children of Israel that whoever kills a soul unless for a soul or for corruption [done] in the land - it is as if he had slain mankind entirely. And whoever saves one - it was if he saved mankind entirely. [5:32]

This clearly echoes a line in the Talmud (there are two separate versions)

Whoever destroys a single soul [Jerusalem Talmud says every soul, I think the Babylonian Talmud says "of Israel" here], it is considered as if he destroyed an entire world. And whoever saves a life, it is considered as if he saved an entire world.

There's also a Talmudic story of Abraham destroying idols as a youth (and claiming to his father that one of the other idols did it) that is not in the Christian tradition, and there broadly similar accounts of Ibrahim/Abraham mocking idols appear in Sura 37 (though I don't think Ibrahim/Abraham actually smashes any of them, he does press one into the water and says, "Drink, drink!"). I believe in both accounts his father is a manufacturer of idols.

(Continued below)