Microsoft: Disable New Mail Desktop Alert for A Specific Account

There is not a direct setting for this in Outlook, but you can achieve this by disabling the generic option to display the New Mail Desktop Alert and then create a rule to display it again when mail arrives via your specified account(s).

<< Disabling New Mail Desktop Alert >>
As said, the first step of the process is to disable the generic option to always display the New Mail Desktop Alert.

** Outlook 2003 and Outlook 2007 **
Tools-> Options…-> button: E-mail Options…-> button: Advanced E-mail Options…-> option: Display a New Mail Desktop Alert

** Outlook 2010 and Outlook 2013 **
File-> Options-> Mail-> section: Message arrival-> option: Display a Desktop Alert

<< Creating a New Mail Desktop Alert rule for specific accounts >>
The next step is to create a rule to display the New Mail Desktop Alert again but only for the accounts that you select.
1.  Open the Rules and Alerts dialog;
** Outlook 2003 and Outlook 2007 **
Tools-> Rules and Alerts… (press OK if you get an HTTP warning)
** Outlook 2010 and Outlook 2013 **
File->  button: Manage Rules & Alerts
2.  When you see the “Apply changes to this folder” drop down list at the top, make sure that the account which you want to exclude is selected here.

3.  Button New Rule…
4.  Select “Start from a blank rule” and verify that “Check messages when they arrive” or “Apply rule on message I receive” is selected.
5.  Press Next to go to the Conditions screen.
6.  Verify that no condition is selected and press Next.
7.  A warning will pop-up stating that this rule will apply to all messages. Press “Yes” to indicate that that is correct.
8.  Select the action “display a Desktop Alert”.
9.  Press Next.
10. Press Finish to complete the rule.


Reference:
Disable New Mail Desktop Alert for specific accounts
http://www.msoutlook.info/question/798