Word is deceptive about this. The tooltip for a link is always *displayed*
as the resolved (ie absolute) reference, even if the stored value is a
relative reference. If the link is entered (on the Insert/Edit Hyerlink
dialog) as a relative value, the displayed absolute value will update
automatically if you move the file. In other words, you needn't do anything.

You can also make use of the 'Hyperlink Base' value on the File > Properties
dialog if you want the links to be relative to something other than the
file's current location.