The expression “to take one’s Bible oath” means: To be absolutely certain; to have no shadow of doubt.
Alluding to the oath that one takes as a witness in a court, in which, under certain circumstances, one places his right hand upon the Bible or holds it, or the New Testament, in his right hand and swears “to tell the truth, the whole truth, and nothing but the truth; so help me God.”
Accordingly, an expression of willingness “to take one’s Bible oath” is a statement that one is willing to enter such a court and to swear such an oath.
Alas, unfortunately the statement is often, if not usually, that of a scoundrel, to whom the sanctity of the Bible or of such an oath is meaningless.
The expression is criminal slang, and usage seems to have started in the past half century.