It’s hard to give a “proof” of the Qur’án, in the sense of convincing others instantly and utterly, but there are a few facts to consider:
Muhammad was illiterate, and is reported to have uttered the chapters of the Qur’án on the spot, noted down by those around him. If you look at the language and complexity of the Qur’án, and how hard it is even for learned individuals to mimic its richness and style, that’s pretty amazing by itself.
Although the Qur’án is relatively short, it became the basis for a civilization that was more successful than the Roman Empire, covered a greater territory, and was reknowned over the centuries for its art, science and philosophy. How can such a small book, spoken aloud by an unlettered man of Arabia during the 7th century, be so rich and complex that it still serves as a guide for millions today?
The Qur’án makes an explicit challenge that if it is not from God, simply imitate it: Write something of equal beauty and power, if you think it’s just the ravings of a poet from the Arabian desert. Although this challenge was given over 1300 years ago, it has nonetheless withstood the test of time.
If you survey the state of the Arabian Peninsula when Muhammad was born, it is shocking to think it could have ever been united by anyone, so afflicted was it by internecine conflicts, prejudices, and barbaric traditions. Compare the result of the Qur’án’s influence on that region, with the conditions it appeared in, and it becomes very hard to reckon it a mere book of poetry. Many men, over millenia, have tried to fashion ideal societies under far more favorable conditions — we still don’t have a living example of Plato’s Republic. Yet the Islamic nations, at their peak, were a light to the world.