How to install a firmware update from an .exe file?

You're not going to be able to install it from the EXE file directly. Although WINE can run some Windows programs, it certainly is not going to support the low-level calls that the EXE would perform to initiate the update.

You have a couple options:

• Many OEMs include a feature in the BIOS which lets the BIOS update itself from the update file on a USB flash drive. You might check to see if your BIOS supports this.
• You might try booting a live Windows PE environment to run the update file from there.
• Install a full copy of Windows 10 temporarily to a USB hard drive, and install from there
• If you're adventurous you could try unpacking the actual BIOS ROM file from the EXE and flashing it directly to the BIOS EMMC block with dd, but I wouldn't really recommend that.