You can't track these natively, but you could build that into the system. I've outlined a small example of how this might be done.

Custom Page:

<!-- Page: Redirect -->
<apex:page controller="redirectController" action="{!redirect}">
  This page is used for page redirections. If you are seeing this message, it means the link was incorrectly configured. Please contact your administrator.
</apex:page>

Custom Controller:

public class RedirectController {
  public ApexPages.PageReference redirect() {
    String Url = ApexPages.currentPage().getParameters().get('url');
    if(url==null || url=='')
      return null;
    ApexPages.PageReference ref = new ApexPages.PageReference(url);
    Tracker__c tracker = new Tracker__c(Url__c=url);
    insert tracker;
    ref.setRedirect(true);
  }
}

Custom Object: Tracker

Name Field: AutoNumber
Custom Field: Url__c (type: URL)

Custom links would be changed (__URL__ is the original URL, escaped, __FRIENDLYNAME__ is the display name):

HYPERLINK( "/apex/Redirect?url=" & __URL__, __FRIENDLYNAME__ )

Hopefully this gives you an idea of how you might implement it.

The object itself would have OwnerId set to the person that clicked the link, Created Date is when they clicked on the link, and URL is the link that was clicked.

Of course, you could make this a lot more complex, like a second custom object to track the types of pages or URLs (to allow rollup summary fields), some other fields to track ownership and assign the records to an administrator (so users can not see/delete the data). I leave such modifications as an exercise for the reader.

Edit: Updated the link to the correct value.