I have two Exchange mailboxes configured in Outlook. One for my email address and another shared one.
UPDATE:
How do I disable desktop new mail alerts for the second mailbox account?
The xxl3ww's answer disables alerts globally. In order to disable email alerts for selected accounts you will need to perform some additional actions. First, disable alerts in Outlook options. Next, create an Outlook rule for each account you want to get alerts from that will run against all received messages (do not specify any conditions) and select Display a Desktop Alert as an action.
File-> Options-> Mail
There is an option under "Message Arrival" for "Display Desktop Alert". Uncheck this box.