The only place in the Torah where the Hebrew word for "axe" appears is in Deut. 19:5. The paragraph is regarding someone who unintentionally killed someone. Such a person is told to flee to a city of refuge where the "blood avenger" is not permitted to kill him. (Without delving into details, this is one of the few situations where the Torah allows intentionally killing another - assuming that the unintentional killer is NOT in the city of refuge.)
One of the examples of unintentional killing described is if a person chops a tree and the steel head of the axe flew off and killed someone.
I don't see any "symbolic" meaning in describing the axe, either here or anywhere else in Tana"ch. Thus, I surmise that "axing a question" is merely a mispronunciation of the word "ask". It's become, IMO an acceptable and tolerated phrase, when I think it should be politely corrected. I can't say how it originated, but, IMO, it seems to be said, mainly, by less-educated people.