No, it's not part of the firmware. It's either defined as a fixed IP-address somewhere in the OS, or the device gets it via DHCP from another device. Usually in home networks a PC connects to a router, and the router gives it an IP-address.
But you might find your modem has a fixed IP-address given by your ISP that it shows to the outside world. For dial-up it changed everytime you made a connection. For ADSL and cable it usually is fixed. Easy to try out by using any of the 'show my IP-address' sites available. That shows it.
To see the internal IP-addresses of your desktop and notebook, either use ipconfig at that device or go into the router management page and find where it tells you about the currently connected devices.
Does the IP address change? I have a router,desktop, notebook, iPad-using applications that connect with an IP Address, iTouch-using iTouch applications that connect with an IP Address. Or is the IP address part of firmware?