I just managed to figure this out myself this is assuming that you have placed an image named favicon.ico in the root/images folder:
Edited Files:
/includes/client/header.inc.php
/includes/staff/header.inc.php
Open client/header.inc.php:
Find:
After add:
Open staff/header.inc.php
Find:
osTicket :: Staff Control Panel
After add:
Save both files.
Refresh.
Not sure why client only requires a ./ and staff requires a ../ but it does so make sure you do that.
Also doesn't really matter where you add the favicon line, so as long as it is located within the header.