To clear all your unread conversation notifications, click on your profile picture or name on the upper right corner of Kona and choose "My Profile." Go to the Notifications tab and click 'Mark all unread conversations as read." Choose "Yes" on the confirmation dialog. Then, click "Continue" to refresh your browser.
(Note: Another way is to hover your mouse over the orange conversation bubble beside the My Kona logo on the top left corner of Kona. Then, hold down the Alt button and click with your mouse. The notifications will clear.)