Yes, if the motherboard manufacturer doesn't provide a way to update the BIOS via Linux you can try using a Windows LiveCD/USB instead. As per my experience though some BIOS updaters do refuse to launch under WinPE (should work fine with Windows To Go but I've never tested this).
However if there is a DOS-based option available I would advise you to use that instead since there's less extraneous stuff involved and thus probably less chance of something interfering with the flashing process. You can follow the instructions here to create a FreeDOS LiveUSB from Linux, or simply use UNetbootin instead:
