To accomplish this you need to first disable notifications for all accounts by:
- Going to File -> Options
- Going to Mail tab
- Unchecking the Display a Desktop Alert
Then you need to enable desktop notifications for each account you want notifications for by creating a rule:
- Under Home tab, click on Rules then click Manage Rules & Alerts...
- Next to Apply changes to this folder:, select the account you DO want to receive notifications for
- Click New Rule... button
- Select Apply this to new messages I receive then click Next
- Click Next again; if a message pop up, click Yes. You should be on the step where it asks "What do you want to do with the message?"
- Check display a Desktop Alert
- Click Finish; if a message pops up, click OK