First things first, you cannot delete your only mailbox. If you just have one mailbox, you will need to create a second mailbox, even if it's just using a fake name and email address. 

Once you have more than one mailbox, go into the Settings > Conversations and select the mailbox you want to delete and scroll down to the bottom of the page. 

There is a section at the bottom with a red background and it contains a big button that says "Delete Mailbox Now". 

⚠️Caution! Before you delete anything, make sure you don't need any of the tickets in your mailbox, because once deleted all of the tickets in the mailbox will be removed as well.