The claim, stated plainly

There is a necessary, non-contingent cause of this universe; it is one, not many; it has communicated with humanity through messengers, most recently and finally through an unlettered seventh-century merchant; and the book he transmitted is that communication, preserved. Every clause is doing work, and each can be probed separately, which is how we'd suggest you take them.

The arguments, with their honest weights

• Contingency and origin: everything in the universe is dependent; a series of dependent things doesn't explain itself, whether the series is short or infinite. The Quran's version is compact: "Were they created from nothing, or are they the creators?" (52:35). This is the classical argument refined by both Muslim philosophers (the kalam cosmological argument is literally named for Islamic theology) and their critics; it establishes, at most, a necessary cause, and honest theists say so.
• Fine-tuning: the physical constants sit in ranges permitting complexity; explanations on offer are design, multiverse, or brute fact. Design is not proven by this; it is, however, no longer the option that gets laughed out of physics departments.
• Consciousness and reason: if your reasoning is exhaustively caused by non-rational particle motion, the confidence you place in that reasoning, including about atheism, needs an account. Islam's claim that mind is not an accident is at least competing on a live problem.
• Morality: you are almost certainly a moral realist at 2am about some things (torturing innocents is wrong, full stop, everywhere). Grounding that "full stop" without something like a transcendent source remains famously hard. Not impossible; hard.

None of these compels belief; arguments rarely compel. Their job is smaller: to show that theism is a rational live option, at which point the specific evidence for a specific revelation becomes worth your time.

The falsifiable bits

Islam, unusually among religions, stakes claims you can check. The Quran challenges doubters to find internal contradiction (4:82) or produce its like (2:23), and claims its own preservation (15:9): manuscript evidence for that last one is summarised, debates included, at how the Quran reached us. The historical footprint of the man, an unlettered trader producing the most studied book in Arabic and a civilisation-founding legal order in 23 years, is examinable through ordinary historiography. And there is a personal experiment with a null hypothesis you control: the tradition's standing suggestion to the sincere undecided is a single honest request, alone, worded however you like: "if there is a God, guide me." Cost: nothing. Reported results: step eight of the seeker's path.

What Islam is not asking of you

Not the abandonment of science (see what this civilisation built and how carefully we suggest reading the "science verses": no overclaiming), not hostility to doubt (the tradition honours the questioner; see what Muslims themselves debate), and not blind leaping: the Quran's constant refrain is "will you not reason?" Its actual demand is the one you already respect: follow the evidence, all of it, including the kinds your current framework files under "unaskable". The hard questions are answered at full strength; start with your best one.