The word "Bible" comes from the Egyptian word for parchment, Byblos. The Greek word biblios means books or scrolls. Later, it was translated into Latin, Biblia, meaning "book."

When the Holy Scriptures took the name: Bible?
In the fifth century, the name Bible began to be given to the entire collection of sacred books, the "Library of Divine Revelation."

Who adopted the name Bible?

The name Bible was adopted by Wickliffe, and came gradually into use in English language.

Also, bible is a book that is considered authoritative in its field e.g. the bible of French cooking.

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The Quran is mentioned 69 times in the Quran.
The word "Quran" comes from the verb "Qara-a" which means "Read".
The "Quran" is the chosen name that is inspired and it is mentioned 69 times in the Quran.
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Example of the verses that mention "Quran":

The Quran contains the best of narratives.

Surah (Chapter) 12:3
In the sending down of this Quran, we will narrate to you (Prophet Mohammad) the best of narratives, of which you were previously unaware.
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The Quran is the guidance and gives the glad tidings to the believers

Surah (Chapter) 17:9
Verily this Quran guides to that which is most upright, and gives the glad tidings to the believers who work deeds of righteousness, that they shall have a magnificent reward;
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All mankind can not produce a book like the Quran!
Surah (Chapter) 17:88
Say: "if the whole of mankind and jinn were to gather together to produce the like of this Quran, they could not produce the like thereof, even if they backed up each other with help and support.
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Whosoever is guided is only guided for himself

Surah (Chapter) 27:92
And to recite the Quran; whosoever is guided is only guided for himself, and to whosoever goes astray, say: 'I am only a Warner. '
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The Quran is easy to understand and remember

Surah (Chapter) 54:17
And we have indeed made the Quran easy to understand and remember: then is there any that will receive admonition?
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The Quran makes things clear; it is mighty, holy, grand, glorious and full of wisdom
To appreciate that, one has to read it.

Surah (Chapter) 15:1
Alif Lam Ra. these are the verses of the book; of a Quran that makes things clear.
Surah (Chapter) 15:87
We have given you the seven dual (verses, al Fatihah) and the mighty holy grand Quran.
Surah (Chapter) 36:2
By the Quran, full of wisdom,-
Surah (Chapter) 50:1
Qaaf, by the glorious Quran!
Surah (Chapter) 56:77
It is indeed a glorious (an honored) Quran,
Surah (Chapter) 85:21
Nay, but it is a glorious Quran.
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If you are hear the recited Quran, listen to it in order to have mercy!

Surah (Chapter) 7:204
When the Quran is read, listen to it with attention, and hold your peace: in order that Allah has mercy upon you.
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Otherwise, are their hearts locked up by them?
Surah (Chapter) 47:24
Do they not then earnestly seek to understand the Quran, or are their hearts locked up by them?
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It follows that the name "Bible" is a man-made and it is not inspired and it is not mentioned in the entire Bible. On the other hand, the name "Quran" is inspired and is mentioned 69 times in the Quran.