1
general::
Disagree with someone, as in John told me Max was sure to win, but I beg to differ? I don't think he has a chance. This courteous formula for expressing disagreement echoes similar uses of beg in the sense of "ask," such as I beg your pardon, so used since about 1600. Also see EXCUSE ME.
American Heritage Idioms
2
general::
Phrase(s): beg to differ (with someone)
Fig. to disagree with someone; to state one’s disagreement with someone in a polite way. (Usually used in a statement made to the person being disagreed with.) • I beg to differ with you, but you have stated everything exactly backwards. • If I may beg to differ, you have not expressed my position as well as you seem to think.
McGrawhill's American Idioms And Phrasal Verbs