The software (Asus Live Update) which downloaded and installed the malicious code was written by ASUS to run on their Windows products. The goal was to automatically look for new versions of programs released on the ASUS website and update the BIOS, drivers, and applications. The update server was compromised, and malicious code was distributed signed with ASUS certificates.
ASUS Live Update was installed by the OEM as part of the Windows OS. ASUS doesn't sell computers with Ubuntu installed from the factory, so anyone running Ubuntu wouldn't have the effected software.
That limits exposure to drivers and BIOS firmwares provided by ASUS. It doesn't appear that ASUS offers Linux specific drivers for their laptops/desktops. A BIOS update though could still be effected if the malicious firmwares made their way into the APT sources.
EDIT: I looked into this a but more and it was the update software itself that was backdoored. The malicious software was hosted on liveupdate01s.asus[.]com and liveupdate01.asus[.]com ASUS update servers. This implies those servers were intended only for the Windows application.