Linux has support for updating various Lenovo models through a piece of software called fwupdmgr. If your laptop is support, it should show up automatically in Gnome Software or similar package managers.
For your laptop, Lenovo has a “Bootable CD” download option for non-Windows users. It’s intended to be written to a CD (but a flash drive will probably also work), for example by using one of those USB DVD drives.
If you don’t have a flash drive for some reason (and I doubt you’ll have a DVD drive in that case), you can try to make the Ubuntu bootloader boot the ISO, though that’s not something for beginners. Here are the official instructions in case you still want to try, but I don’t think I’d bother.
The easiest method may be to contact Lenovo and ask them how to do it. I think they’ll refer you to the bootable ISO. If they don’t make their updates available for anything but Linux, you’re going to have an annoying time.
Spending the five dollars on a flash drive to write the bootable CD to would be worth it in my opinion.
To answer your question: if the software manager doesn’t offer you the firmware update already, the easiest (not necessarily easy) way would be manually adding a bootloader entry to your Grub configuration to boot the update ISO you can download from Lenovos’s website.
The second easiest way would probably be to extract the firmware updater from either the Windows download or the ISO file, extracting the .efi files and the .rom files, placing them on your EFI partition, and using the boot menu to manually boot the firmware updater.
Or, to answer more succinctly: if you don’t get those updates already, there’s no easy way without a bootable medium. Sorry. Tell Lenovo to publish the firmware updates through the standard Linux channels like they do for other laptops.