Depending on your setup, users will at times receive an authentication prompt when opening Microsoft office documents even when the site is in the Trusted Sites list or the Intranet sites list. There are reasons for this which are complex but you may have a good read of the KB article:

The fix is to modify the SharePoint web application config file (web.config) in the app directory to specifically bypass the authentication prompts. As a rule of thumb, please backup your web.config before making any changes.