Will Mac OSX 10.5.4 or 10.5.8 run on a PowerMac g5 quad?
According to Mactracker, the latest version of Mac OS X any version of an Apple Power Mac G5 Desktop can take is Mac OS X 10.5.8 although Mac OS X 10.5.4 will also run on them.

The information I presented was what is reported by Mactracker and they usually are correct with the information they provide.  In any case all sources I've looked at for the Power Mac G5 Quad 2.5GHz M9592LL/A PowerMac11,2 the only quad core Power Mac G5 produced all say the same thing, latest or maximum OS is "Mac OS X 10.5.8".  Unfourtunatly Apple's own page Power Mac G5 (Late 2005) - Technical Specifications does not say. So I guess the answer should be contact Apple via phone or stop by an Apple store for what would be considered an authoritative source.