Islam has no symbols. Quran doesn't suggest or order to use any symbol, but doesn't prohibit as well.
The story behind the reason why some people use the crescent moon as a symbol is explained in the passage below.
The crescent moon and star symbol actually pre-dates Islam by several thousand years. Information on the origins of the symbol are difficult to ascertain, but most sources agree that these ancient celestial symbols were in use by the peoples of Central Asia and Siberia in their worship of sun, moon, and sky gods. There are also reports that the crescent moon and star were used to represent the Carthaginian goddess Tanit or the Greek goddess Diana.
The city of Byzantium (later known as Constantinople and Istanbul) adopted the crescent moon as its symbol. According to some reports, they chose it in honor of the goddess Diana. Others indicate that it dates back to a battle in which the Romans defeated the Goths on the first day of a lunar month. In any event, the crescent moon was featured on the city's flag even before the birth of Christ.
The early Muslim community did not really have a symbol. During the time of the Prophet Muhammad (peace be upon him), Islamic armies and caravans flew simple solid-colored flags (generally black, green, or white) for identification purposes. In later generations, the Muslim leaders continued to use a simple black, white, or green flag with no markings, writing, or symbolism on it.
It wasn't until the Ottoman Empire that the crescent moon and star became affiliated with the Muslim world. When the Turks conquered Constantinople (Istanbul) in 1453, they adopted the city's existing flag and symbol. Legend holds that the founder of the Ottoman Empire, Osman, had a dream in which the crescent moon stretched from one end of the earth to the other. Taking this as a good omen, he chose to keep the crescent and make it the symbol of his dynasty. There is speculation that the five points on the star represent the five pillars of Islam, but this is pure conjecture. The five points were not standard on the Ottoman flags, and as you will see on the following page, it is still not standard on flags used in the Muslim world today.
For hundreds of years, the Ottoman Empire ruled over the Muslim world. After centuries of battle with Christian Europe, it is understandable how the symbols of this empire became linked in people's minds with the faith of Islam as a whole.
Based on this history, many Muslims reject using the crescent moon as a symbol of Islam. The faith of Islam has historically had no symbol, and many refuse to accept what is essentially an ancient pagan icon. It is certainly not in uniform use among Muslims.