I highly doubt it. Usually the Macs can only run the operating systems that came with them or greater. There are some "transitional" Macs that originally ran an older version and were then "refreshed" with a newly released OS from Apple when they were sold, but beyond that no.

The only way you would be able to run System 7.6 on a stock G4 Mac would be through emulation either through Basilisk or through SheepShaver (these I believe are mainly available in OS X).