This might sound silly, but does your Mini have modem board installed? If I'm not mistaken, modem was offered either as 'built to order' or as an upgrade kit.
That is a pertinent question, although I can say from my personal experience of having owned (and still owning one) G4 mini that, from what I could see, the modem socket and card would come with all "first gen" Mac minis (1.25GHz and 1.42GHz), whereas the "silent upgrade" ones (1.33GHz and 1.5GHz) would come with
both lacking, and the modem socket would be closed off with a plastic, grey square that fits the hole. (Well, to be absolutely accurate, this all could have been coincidental, and I cannot be certain about the 1.33GHz ones, as I never owned those.)
Incidentally, I tried transplantling the modem card
and socket from a first gen mini to a silent upgrade one, and then the silent upgrade one simply
died, even after removing the modem parts I installed afterwards. No idea how that happened, as I thought I was careful with static electricity, fitting etc. but who knows.