How can I update modem firmware?

It doesn't matter at all if you own the cable modem or not. Providers have policies (for several good reasons) about how endpoints are managed and most of them (especially the big ones, Comcast, Cox, Charter/TWC/Spectrum, etc) require that in order to connect to their network, your cable modem must run their firmware. It's not even a choice, during initial connection handshake the cable modem downloads the appropriate firmware (every time, no matter what) and runs it. Trying to update the firmware to some other version, if even successful, would only result in losing connectivity until you reboot the modem, at which time it will load again from the network.

Now, to update modem firmware, it's important to not confuse the cable modem with the router/firewall gateway device, in most circumstances these are two different pieces (the modem is strictly responsible for providing IP/ethernet connectivity over the coax, nothing more) so a much different answer would apply if you ask the same question about your router/firewall device.