Source [Disable New Mail Desktop Alert for specific accounts][1]

> 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
>
> - Uncheck Display a Desktop Alert 

> ![enter image description here][2]

> **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.
> 
> The approach is slightly different for when you want to disable alerts
> for only 1 account or multiple.
> 
> **Disable alert for 1 specific account**
> 
> When you have multiple accounts configured and only want to disable
> the alerts for 1 specific account, then you need to create the
> following rule:
> 
>  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.
> 
>     ![enter image description here][3]
> 
> 
> 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.   Select: except through the specified account
> 11.    At the bottom, click on “specified” and then select the account for which you want to see the alert.
> 12.    Press Finish to complete the rule.
> 
> The entire rule will now read as follows:
> 
>     Apply this rule after the message arrives
>     on this computer only
>     display a Desktop Alert
>     except through the <account name> account
> 
> For the other email accounts which still need alerts:
> 
> - Repeat steps 1 to 12 **except for 10 and 11** When you had the “Apply changes to this folder” drop down list as mentioned in step 2,
> you’ll need to repeat the above steps for all the other accounts
> listed but you can then skip step 10 and 11. 
> 
>   If you do not do this, you won’t see alerts for these accounts.

---

Source [New mail envelope and Windows 7][4]

>Outlook 2010

>When you are using Outlook 2010 on Windows 7, then the new mail envelope will no longer be displayed in the Notification Area. 
> 
> Instead, the icon of Outlook on the Taskbar will actually change to
> inform you about a new message. 
> 
> Unfortunately, when you’ve set your Taskbar in Windows 7 to use small
> icons, you will not get an envelope notification at all. 
> 
> In that case you could consider reconfiguring the New Mail Desktop
> Alert to be displayed as a permanent notification.
> 
> Note 1: 
> 
> The new mail envelope is only being displayed for messages delivered
> to the (default) Inbox. For additional accounts you can use a New Mail
> Desktop Alert.
> 
> Note 2: 
> 
> The option to show/hide the new mail envelope for Outlook can be found
> in;
> 
> -    Outlook 2007 and previous
>  
>      Tools-> Options…-> tab Preferences-> button E-mail Options…-> button Advanced E-mail Options…-> option: Show an envelope in the
> notification area
> 
> -    Outlook 2010
>  
>      File-> section Options-> section Mail-> option group Message arrival-> option: Show an envelope in the taskbar



  [1]: http://www.msoutlook.info/question/disable-new-mail-alert-for-specific-accounts
  [2]: https://i.sstatic.net/vfVmV.png
  [3]: https://i.sstatic.net/umbQk.png
  [4]: http://www.msoutlook.info/question/398