There are two reasons why you don't see new incoming visitor posts in the Inbox:

You have completely disabled the option to post to your page in Page settings on Facebook.

In this case, no posts can be posted on your page, therefore there is nothing we can show you in Inbox.

You require the visitor posts to be reviewed by you before they are visible on your Facebook page.

In this case, posts are actually coming into the Inbox, but they're all being automatically hidden and won't appear in the To review filter (as all new incoming items do). By hiding them, we actually review them, so they'll appear in All filter (as reviewed and hidden items).