Indeed, in the given context the phrase "whether or not" need not be translated; for, with respect to the sentence "I asked him whether or not he had met Mark", to say "我問他認不認識 Mark" suffices. And this is an idiomatic way to "translate".
In general the phrase "whether or not" is not to be one-one translated. For instance, the sentence "I do not know whether or not it will rain tomorrow" corresponds in Chinese the sentence "我不知道明天會不會下雨" or less idiomatically "我不知道明天下雨不下雨". The general rule is obvious now:
Let S be a true subject, let V be a verb, and let O be an object for which "whether or not S + V + O" is considered. If idiomaticness is not rigidly required, and if V' is the Chinese counterpart of V, then for "whether or not S + V + O" to be translated in Chinese it suffices to say "S + V'不V' + O".
Therefore, to translate in Chinese the given sentence "whether or not he had met Mark", where we have replaced S = he and V = had met and V' = 認識 and O = Mark, by the rule above we have
S + V'不V' + O = 他 認識不認識 Mark.
I hope these helpful.