(1) She had done what God asked of her.
(2) I inquired of them the score.
(Examples are from COCA)
I can guess from NED (III.10.b) that when you use ‘ask’ or ‘inquire,’ construed with ‘of,’ you should regard that something you ask or inquire are acquired from of’s complement. For example (2), you expected the score would come out of them. But the trace, in the past you used ‘at’ instead of ‘of’ for both ask and inquire, makes you think of-complement as both your seeking orientation and whence the result comes. Is this how you think when you use ‘ask’ or ‘inquire’?
I inquired of them the score would be regarded as stilted today in most sections of the US, with the possible exception of the so-called "Deep South". It is a stereotypical "southern country gentleman" manner of speaking. See the Google ngram for (inquire of) English 1700-2000, and the same for American English. That form has been for the most part replaced with "ask". I asked them the score. or I asked them what the score was.
What are you asking of me? usually means "What are you asking me to do?" not "What piece of information are you seeking?" When something is "asked of" someone, it typically involves their taking action of considerable importance.