Facebook only allows you to delete conversations one by one individually. There is no option to clear all conversations at once but there is a Google Chrome extension that can easily clear your entire inbox.
Facebook – Delete All Messages allows you to clear all Facebook messages at once from you Facebook messages section.
Start by adding it in your Google Chrome or Chromium browser, after installation it adds a new page section in Chrome’s address bar.
After adding the extension, visit your Facebook messages page and click on the extension’s icon. Now from the popup bar click the ‘Launch’ button and that’s all.