If you go back to the other computer and open the document there, the links should update automatically, shouldn't they? There are some settings in the Preferences, in the File Handling tab. Maybe any of thoose are useful to you.
If you save your InDesign files and their assets into your user > Creative Cloud Files folder they will automatically sync to your subscription’s cloud account. If you log into the second computer using the same account the files will also sync to that computer.