QGIS doesn't have a built-in way to select close-matching fonts if it finds a missing font.
Hence you need to have all fonts (and any other external resources) used available on all systems where you want the project to work.
That means either installing Arial on your Linux systems, and specifying that in your project (knowing its fairly safe that it will be on your Windows systems). Or installing Liberation on your Windows machines and specifying that assuming it will be on your Linux boxes.
Arial can be downloaded via this link, read the license first:
https://www.freedesktop.org/software/fontconfig/webfonts/
As an openly licensed font, Liberation Sans can be downloaded from most font archive sites.