Under Options, go to the Mail tab. Disable the "Display a Desktop Alert" option. Create a new rule to alert you about email received for a specific account; choose the "through the specified account" in the criteria and "display a Desktop Alert" in the actions.

If multiple accounts are set up in your Outlook application, Microsoft has not provided a direct setting to disable desktop alerts for individual accounts. However, you may disable alerts for all accounts and then create a custom rule to alert you about new email received for specific accounts. This has the same end result.

1) Disable desktop alerts for all accounts.

2) Create a rule to alert you about new email received for specific accounts.

OUTCOME

The new mail desktop alerts will be disabled in Outlook except for the account(s) that you specify.