The books, with translations in French and English, were handed out in the capital Paris by a group of young members of the Associations (UAM93).
They also included books introducing the tenets and teachings of Islam, according to al-Khabar website.
The UEFA European Championship 2016 is a month-long event that has drawn about seven million fans.
Muhammad Hanish, secretary general of the union, said the unprecedented measure is aimed at introducing Islam and dispelling myths and distorted image of the religion.
He said fans from different European countries will go home with these copies of the Quran and books to learn about Islam.
He noted that recent terrorist attacks by some so-called Muslims has tarnished the image of Islam and led to increasing Islamophobia in France and other European countries.
A recent poll showed that Islamophobia has risen sharply in the two largest Eurozone countries, France and Germany.
Hanish said the UAM93 seeks to promote the image of Islam through cultural activities and counter radical moves and propaganda spread against Islam via the cyberspace.