Had you stored your QAT (Quick Access Toolbar) modifications in a global template rather than using Word's default, it would be. It still may be.
See,
Modifying the Quick Access Toolbar (QAT) in Microsoft Word
Note that copying a normal template from one system to another is not generally a good idea. If they are not the same version of Word, you can get everything in "compatibility mode."
Most of my customizations are in some separate files that easily transfer both from one computer to another and from one version of Word to another.