Panasonic Gamecube Q - error in GC startup

Bit late to the party here. Unfortunately I don't recall seeing an internal battery when I've taken the Q apart, so can't be of much help

 

You can get the same symptoms if you disconnect the cable going from a GameCube's controller/memory card PCB. The CR2032 is mounted there and if you pull the ribbon the time/date settings will get wiped and you won't be able to control it.

I don't know anything about how the Q is setup internally but it seems like something has damaged communication between the two sets of components.