"He inquired of me that I be quiet"
Does this sentence make sense?
Is it possible to use that clause as a object of the verb inquire?
You could write 'He requested that I be quiet'; but if you said it you could expect the response 'Ooh, get you - who's swallowed Strunk and White then?!' The normal way to say it would be something like 'He asked me to be quiet/to shut up/ to shoosh/to keep mum...' (Maybe not the last, which is rather dated; it persists mostly in the semi-fossilized 'Mum's the word' [='Be quiet/Hush/Can it/Zip it...].