1. If the font is not available, both on Linux and Windows an effort is made to replace it with a font from the same "font family". This can lead to disappointing results, though.
2. A better way is to include the font as a font resource, so the font is part of your application.
I often use 1), but if the font is really special, say, e.g. for music notation, I always use 2). In commercial software I only use 2) except if I use a font that is known to be part of the OS distribution and the application is restricted to that OS.
When in doubt, or X--platform, choose 2)
Adding a font resource is quite easy and I already gave an example sometime ago.
Note that many fonts are copyrighted or otherwise subject to licensing, so try to use a font without many restrictions, like "Creative Commons" versions. Many of the best fonts have such a license.
Of course if you have an interest in Typography/Typesetting you can also design your own font. There are many freeware font designers available. (Challenging, though, at least to me)