There is no graphics “card” per se in the G4 minis.
It’s an on-board situation that came w/ Radeon 9200.
Either with 32 or 64 MB of VRAM.
(The 1.5 GHz G4 mini has 64 MB.)
Many HDMI adapters are finicky at best.
With these minis, perhaps even more so.
Skip the Philips 275E unless it also provides
DVI or VGA (with adapter dongle) connections.
But if it does… I’d still test a different monitor.
Now, anyone successfully running HDMI
…to such a monitor should chime in here.
I use many varieties of DVI & VGA monitors
here with various G4 minis. No problems.