When the Book of Mormon was written, the original writers had access to a Hebrew record referred to as 'The Brass Plates.' These plates contained "...the five books of Moses,which gave an account of the creation of the world, and also of 
Adam and Eve, who were our first parents; And also a record of the Jews from the beginning, even down 
to the commencement of the reign of Zedekiah, king of Judah;And also the prophecies of the holy prophets, from the 
beginning, even down to the commencement of the reign of 
Zedekiah; and also many prophecies which have been spoken by the 
mouth of Jeremiah." (1 Nephi 5:11 -13) 
This record, as described, then, contains about sixty percent of the Old testament Bible as we know it, with a more complete history and record than we currently have in the Bible today.  
Another note is that, amongst the material quoted from this record, there are numerous differences from our current biblical writings; the Book of Mormon writers had access to (more or less)the "original documents" from which the Bible was compiled, and their writings were not marred by the numerous revisions and re-translations that are apparent in the KJV today. 
To answer your question directly: the Book of Mormon does not quote directly from the KJV; rather the material in question simply derives from the same source- the Jewish records and histories that constitute the Old Testament (up until 600 BC that is).