Why should they? He was not presenting those stories in the Qur'an as new stuff, previously unheard of. On the contrary, the Biblical content of the Qur'an has often what seems a very allusive tone, suggesting that the intended audience was assumed to be broadly familiar with the characters and topics referred to, and only needed to set the existing record straight. In some cases, exegesis points to exchanges with Jewish opponents, who of course would know about Biblical patriarchs. At other times, one sees the Qur'anic text saying that people were asking questions about earlier prophets. In general, it seems plausible that Arabia had monotheists of various kinds and some sort of knowledge about earlier Abrahimitic scripture was relatively common, even though perhaps not prevalent. It is hard to say much definitive about the religious picture of the time: the Muslim tradition is a mine of information, but sometimes inconsistent and needs to be taken with a grain of salt. Radical revisionists miss the point, and the trend of evidence is not making a lot of favors to their main arguments. External sources are useful, but not very informative and they contain their own biases and inconsistencies.
The Qur'an rarely seems to feel much need to explain from the ground up who Musa, Nuh, Ibrahim or Jesus were, just to clarify points about their mission. To be fair, however, Qur'anic style is more allusive, as opposed to explanatory, in general.
The Muslim tradition on the life of the Prophet mentions a relative of his wife who was very well-versed in Christian Scripture, who would have confirmed his prophecy. It may be a hagiographical topos of course, but confirms that Muslims a century after the Prophet considered knowledge of Christianity in the Prophet's Mecca a credible situation.
