Sure - the quest menu part's Quest Box Settings have options for which states to add (e.g. Add Active, Add Inactive, etc.).
For such a setup, set up multiple menu screens, one for each state you want to separately display, and use a Menu List menu part for switching between them. A possible setup could be:
- Screen A: Menu List part, this is the screen you call and has the other screens added as menu items
- Screen B: Quest part with Add Active enabled
- Screen C: Quest part with Add Finished enabled
- Screen D: Quest part with Add Failed enabled