One should perhaps not rule out that the internal video circuits are faulty in a way that a proper video output is not possible. Even file sharing via Ethernet would be difficult without a display.
With access to another Mac with a FireWire port, you may want to try target disk mode (Use target disk mode to share files between two Mac computers - Apple Support). The PowerBook G4 (there are several models, all with FireWire) could then be used as an external hard drive for that second computer. Should be OK without a working PowerBook display.
If nothing else works, you could remove the PowerBook's internal hard drive, and then connect it to another computer via a universal USB hard drive adapter (one example at http://www.newertech.com/products/usb3_universaldriveadap.php; similar devices can be found in computer shops).