An external hard disk drive enclosure with FireWire400/800 ports, with others
may be able to be used; but not as an adapter from one kind of port to another.
The old iMac uses USB, perhaps as slow as 1.1; that won't work with newer
USB3.0; but both the newer and older computers could use FireWire; or maybe
ethernet for computer to computer.
Newest models have Thunderbolt data ports, so adapter cables are needed.
And then you'll need cables to match whatever combination you end up with.
{You may have tried something or another, but not posted back to say if you did.
And this thread is nearly two weeks old, now. I just happened to find it here.}
For what you suggest, you may be able to find a suitable cable, perhaps FireWire
or Ethernet, depending on the ports available and cables still applicable to both.
Does newer iMac have FireWire ports? You could buy a FW400 to FW800 cable where
the ends are different, if the newer machine has FireWire800. [The old one has FW400.]
This would have to be equipped with the protruding connectors so each could make
the connection between each computer's built-in ports. Not all cable sets do this.
This would be the fastest way to copy content between machines, if FW 800 in newer mac.
Port-powered external hard drives may be under-powered and fail to work, so that is why
I suggest enclosed HDD with its own AC power adapter that will not rely on FW or USB
for the power to run the drive and the electronics inside the enclosure.
example of external enclosures with multiple ports and its own power adapter:
http://eshop.macsales.com/shop/firewire/1394/USB/EliteAL/eSATA_FW800_FW400_USB
This is an image of the lesser model series with fewer ports. The FW800 and USB3.0
are both backward compatible to their older counterparts, FW400 & USB2.0. But even
so the old 1GHz iMac G4 with USB1.1, would not be able to use newer USB3.0; yet
should be able to connect via FireWire400 to FireWire800 by special cable.
If the iMac G4 is still working, that's a plus. Attempting to remove the old IDE/ATA HDD
and access it outside the PowerPC iMacG4, is really a troublesome effort, one to avoid.
But the PPC iMacG4 can be started in Target Disk Mode and accessed as a hard drive
via FireWire cable, too. Then files can be seen as though the old Mac is an external HD.
In any event...
Good luck & happy computing! 🙂