There are about 1.6 billion Muslims, every one of whom has at least one Quran, so the total figure is probably over 2 billion.
There are no official records of total number of copies of Holy Quran which is a word of God like Bible.
Every Muslim has more than one copy of Quran at home. At the same time, there are millions of mosques which have millions of copies of Quran.
So, there are billions of copies of Quran because population of Muslims in the world is 1.8 billions.
Secondly, there are millions of Muslims who have learned Quran by heart.