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As you have initially pointed out, the main and original difference between Christians and Muslim is that Christians redoubt the inspired nature of the Quran.
Nevertheless, there is a group/movement/denomination that may be counted among Christians that partially accepts the Quran as inspired, along with other religious teachings but who see the capability of humans to receive revelation submitted to human limitations. This is the Universal Unitarian movement, See https://en.wikipedia.org/wiki/Unitarian_Universalism?wprov=sfla1.
For the syncretism of the philosophy or religion, it's not unanimously seen as a Christian denomination, and their acceptance of the Quran is certainly not the way as in Islam.
I do not know any Christian church that accepts the Quran as an inspired writing.
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Does any Christian denomination consider the Quran as inspired?
The short answer is no.
Although certain truths may be found in all major religions, including Islam, no Christian denominations believe the Quran (Koran) as being inspired (from God).
Christians should have a respect for Quran and the beliefs of Muslims, but do not compromise Christian beliefs that the Quran is is inspired from God. From a Christian point of view it is not.
Catholics, for example do not believe that the Quran is inspired, yet acknowledges that there are some truths within Islamic faith or other non-Christian religion, for that matter.
The Catholic Church rejects nothing that is true and holy in these religions. - Nostra Aetate
Nevertheless, the Catholic Encyclopedia explains the six sources from which the Quran seems to be based on.
The sources of the Koran be reduced to six:
The Old Testament (canonical and apocryphal) and the hybrid Judaism of the late rabbinical schools. During Mohammed's time the Jews were numerous in many parts of Arabia, especially around Medina. Familiarity with them is undoubtly responsible for many Old Testament stories alluded to in Koran. Later Judaism and Rabbinism are equally well represented (Geiger, "Was hat Mohammed aus dem Judenthum aufgenommen?", Wiesbaden, 1833; tr. "Judaism and Islam", Madras, 1898).
The New Testament (canonical and apocryphal) and various heretical doctrines. On his journeys between Syria, Hijaz, and Yemen, Mohammed had every opportunity to come in close touch with Yemenite, Abyssinian, Ghassanite, and Syrian Christians, especially heretic. Hence, while the influence of orthodox Christianity upon the Koran has been slight, apocryphal and heretical Christian legends, on the other hand, are one of the original sources of Koranic faith. (See Muir, op. cit. infra, 66-239; Tisdall, "The Original Sources of the Qur'an", London, 1905, 55-211.)
Sabaism, a combination of Judaism, Manicheism, and old disfigured Babylonian heathenism.
Zoroastrianism. On account of Persia's political influence in the north-eastern part of Arabia, it is natural to find Zoroastrian elements in the Koran.
Hanifism, the adherents of which, called Hanifs, must have been considerable in number and influence, as it is known from contemporary Arabian sources that twelve of Mohammed's followers were members of this sect.
Native ancient and contemporary Arabian heathen beliefs and practices. Wellhausen has collected in his "Reste des arabischen Heidentums" (Berlin, 1897) all that is known of pre-Islamic Arabian heathen belief, traditions, customs, and superstitions, many of which are either alluded to or accepted and incorporated in the Koran. From the various sects and creeds, and Abul-Fida, the well-known historian and geographer of the twelfth century, it is clear that religious beliefs and practices of the Arabs of Mohammed's day form one of the many sources of Islam. From this heathen source Islam derived the practices of polygamy and slavery, which Mohammed sanctioned by adopting them.
Other Christian sources may be perused at oneβs leisure in the following articles: