Hi,
It would probably be correct to assume that the iBook G4 in question has the (optional) AirPort Extreme card (802.11b/g) installed.
Also, which exact Mac operating system version?
Does the computer see the Wi-Fi network provided by the router?
What kind of error message?
WPA security should be OK, but WPA2 may not work. Please check this (if you have an old/spare router, you could use this one for an experimental local Wi-Fi network setup with a temporary, lower security).
If there is a problem with modern Wi-Fi security, one solution could be to connect a Wi-Fi range extender with wireless Ethernet bridge capabilities (an Edimax EW-7438RPn Mini operating in a bridge/client/adapter mode would be just one example) to the Ethernet port of the iBook G4.
Alternatively, you could, of course, test a wired (Ethernet) connection to one of the LAN ports of a Wi-Fi router in order to gain access to the Internet that way.
Also, please note that the early web browsers compatible with this Mac cannot handle modern secure (https) web sites. The remaining plain http sites should work well, though. If you have Mac OS X 10.4 Tiger, it may be possible to use the now discontinued TenFourFox browser even with a few secure web pages.
https://everymac.com/systems/apple/ibook/specs/ibook_g4_800_12.html
https://everymac.com/systems/apple/ibook/specs/ibook_g4_933_14.html
https://everymac.com/systems/apple/ibook/specs/ibook_g4_1.0_14.html