There is a possibility the display adapter may not be the correct model.
Or that it could be inoperative. Some versions look similar but are not...
Also, the external display has to be one supported by the iMac G4 output.
Mini VGA, to standard VGA, is the primary path.
see search:
https://www.google.com/?gws_rd=ssl#q=apple+vga+display+adapter+imac+g4
Did you restart the computer with the external display attached & turned on?
Sometimes after such an event the extra display will appear in Displays
preference panel.
Depending on the system version running in the computer, the Preferences
panel will also vary; so the answer would to some extent, as well.
Though I have three iMac G4 (17-inch 1.25GHz) computers, none are up
and running; of those one still can, but needs an optical drive. And maybe
by now, also a 3.6v 1/2AA lithium PRAM battery replaced, too.
In the older supported OS X and OS 9 versions (early iMacG4 dual-boots)
the systems are so different as to make precision troubleshooting methods
rather hit-and-miss; usually miss.
The external display was proven to work with a different computer, and it is
a VGA, not a DVI? There was a third-party software that could be used to
help control an external display, and some users found them helpful to set
up a display with older iMac G4, iBook G4 or PowerBook, with external.
•Switch ResX
http://www.madrau.com/index.html
older version download (shareware) here:
http://www.madrau.com/srx_download/download.html
Then, there was this screen-span utility that was not officially supported
but worked with older computers, to help them with external displays:
http://www.rutemoeller.com/mp/ibook/ibook_e.html & it may be a gamble.
These suggest to add extra ability to change or allow effects to an additional
display; usually if you have working adapters correct to the display & the port
your computer is equipped with, a restart will allow control preferences to work.
Some of the later VGA adapters for MacBook look like the earlier VGA ones
for iBook/PowerBook/iMac G4; so be aware they may look similar but aren't.
Mini VGA to VGA adapter versions can vary, too. I have an old unmarked one
that works with iBook G4 and iMac G4 series; bought second-hand.
Not sure if this will help you much...
Good luck & happy computing! 🙂