There is no Christian bible. What we call the Old testament is the Tanach (Torah, Neviim, and Ktuvim). The sections that mention King David are in the Book of Samuel (among others) which are part of Ktuvim (writings) – and that predated Christ by hundreds of years.
The so called Christian bible is sometimes referred to as the New Testament.
Maybe you are referring to the King James (or other later) translations of the Tanach, which were written with a decidedly Christian spin 1400 years after they were originally released.
But the David section is very definitely, and in no way Christian whatsoever.
But