The Delete Zone icon becomes available when you have selected a zone by single clicking on it.

When in Zone View mode, this will bring up a Delete Zone window with a series of checkboxes for every server that serves the zone. By clicking the checkboxes by each server, QuickDNS Manager will allow you to remove the zone from any of your QuickDNS servers where the zone is hosted.

When in Server View mode, deleting a zone will delete it only on the associated server. A confirmation window will appear to make sure you intend to delete the zone from that server.