Hello,
For BIOS updates Lenovo publishes ALL models with a Windows Executable file. So with Windows operating system active, I download the .exe file, save it to my Windows 10 desktop, close ALL other applications using the "X" in the corner of the window, then Right Click on the BIOS update executable select Run as Administrator, then follow the instructions within the popup boxes. Including usually a shutdown and reboot.
So if you made a dual boot installation of Windows and your distribution of Linux, simply boot into windows and perform these tasks. If you don't have Windows installed there are additional ways:
On the download page for the BIOS update SOME models have an alternate link to download a bootable USB image to use for updating BIOS.
Check the README file within your .exe BIOS distribution, often this is available separately on the Lenovo Download page. SOME of those contain instruction on how to extract and create a bootable USB bios installation stick from within the .exe file.
Under linux, use your distribution's package management tool to download the linux package fwupd. Try command "man fwupdmgr" first, it might already be installed. After installing from a command line issue linux command "fwupdmgr refresh". IF bios is available it will provide a message it is available. Then following further instructions on the man pages for fwupdmgr issue the commands needed to actually download the bios update and install it. Likely superuser permissions will be necessary. If a bios is not available from fwupdmgr today, check again in a few weeks. More and more systems are being enabled over time, however universal coverage for all models of Lenovo computers is not planned.
These are the only Lenovo supported ways to access and install bios.
Still other ways have been described, your mileage will vary.
Sincerely,
Usedtoberich