The mechanics of the atonement have been explored in prior questions, so here the only pertinent issues are those relating to the Trinity and the status of the Qur’an. With respect to the former, the claim of the Qur’an is belied by the fact that it purports to provide a path to God’s forgiveness apart from an atonement of any sort. Given that the nature of God can be shown to require the punishment of sinners (as demonstrated elsewhere), the Qur’an’s presupposition to the contrary must be false (and thus, it cannot be the word of God).
With respect to the latter, it is not the case that the Trinity claims that God is one and three of precisely the same thing; rather, he is one being and three distinct persons. Thus, more is required if one wishes to show the concept to be contradictory.