Yes, in answer to your questions, there is proof in the Qur'an itself that no human could have wrote it. First of all, if the Book itself contains knowledge about Allah which go far beyond what a human can know, then that is another proof for its truthfulness. Secondly, it must have distinguishable signs which someone would have to see with their own eyes as a proof of its truthfulness, this could be what happened in the past, and what happens now as well as the future. Thirdly, does its statements support each other? By this I mean, does it confirm its own truthfulness?
The first thing to note is that the Holy Qur'an is a book which contains statements clearly identifying it as an address from Allah (4:163, 4:166, 9:97, 18:110, 20:4, 20:114, 21:45, 26:192, 32:2, 36:5, 39:1, 40:2, 41:2, 41:42, 42:52, 45:2, 46:2, 53:4, 56:80, 69:43, 72:23)
As for the knowledge about Allah, it has the most exquisite verses which explain the Divine attributes. No other religion has explained the perfect attributes of Allah. It explains that He never dies, never sleeps, He is the King of all kings, He does not have a son, He has never been conceived Himself, He is the First and Last, with no one and nothing before Him or after Him, He is Time itself and He is beyond comprehension. We only know Him by the opposites mentioned in His attributes, otherwise there is no way to understand His attributes.
And in short:
"There is nothing like unto Him." (42:11)
All other religions fall short of explaining Allah as the Perfect Being. You can't tell me of any other religion where God is One entirely, and that the explanation of His attributed negates any other partners to be attributed alongside Him. Islam removes all idol-worship in its teachings.
There are many statements which the Holy Qur'an has which could not have been conceived of in the time of the Arabs. But just one example suffices, the prophecy in which the Qur'an said that when the Romans had been defeated by the Persians, but after 3-9 years (Bidh'a, interpreted in Hadith traditions as such), the Romans would push back against the Persians. This future occurrence during that time seemed an unfounded statement, since the Romans were seen as too weak to ever regain their loss. But they did eventually, and it was miraculous. No one could have seen that coming. The book "Rise and Fall of the Roman Empire" by Edward Gibbons even records how the prophecy of Prophet Muhammad (saw), which was stated in the Holy Qur'an, came to pass as he had said. He could not have said this on his won, it was a strong proof for this authenticity as a Prophet of God.
But this is just one instance of his being truthful. The Qur'an has a lot of other examples, many of them would take too much time to sift through and find. But this is a very well-known one, and it was even attested to by a famous historian such as Edward Gibbons, who has testified to the extraordinary prophecy he made coming true.
As for scientific proofs from the Qur'an, one which stands out from all of the rest, is the one in which the Holy Qur'an says that Allah widens the expanse of the Heavens by His Power:
51:47 And We constructed the Heaven with Strength (Power), and indeed We surely are its Expanders.
(The translation I gave is a paraphrase of the word-for-word detailed translation from the following page:
http://corpus.quran.com/wordbyword.jsp?chapter=51&verse=47)
The verb la musi'oon indicates the action being continuous and constantly occurring. Edward Hubble found out that the universe was expanding in the 20th century. The Qur'an was revealed in 7th century Arabia, 13 centuries earlier. No one could have conceived of this. The universe was considered fixed and the earth at its center, fixed as well.
For me, just this one instance of the Qur'ans description of the universe is enough to satisfy me that it is a Divine source. How could anyone have said that, and other statements which such as the earth's formation and other heavenly bodies from smoke (nebulas)? Furthermore, the Qur'an said the earth was round, because the moon and sun were said to float in orbits, (falak). The term falak refers to a rounded motion, like around an orb. So if you have a ball, then the sun and moon travel right around the shape of a ball. That describes the type of orbital path these heavenly bodies make around earth.
