Dear Tracey_150,
Good day!
Thank you for posting to Microsoft Community. We are glad to assist.
As per your mentioned description about "Desktop Notification Do Not Show". To understand the situation and be able to offer you relevant suggestions, we would need a little more information from you. Kindly help us in providing the following information.
May I please know if you are using Microsoft 365 Exchange email account or Microsoft personal account (such as Outlook.com or hotmail.com)?
May I please confirm that Display a Desktop Alert is selected? Turn new message alert pop-up on or off - Microsoft Support To check this kindly Select File > Options > Mail-->Under Message arrival, select or clear the Display a Desktop Alert check box and then select OK.

Note: To suppress or enable other notifications such as sounds, mouse pointer changes, or displaying an envelope icon in the taskbar, check or clear the respective check boxes for those functions.
May I please confirm that you have enabled notifications for Outlook? To check this Kindly Search Settings in the search bar of Windows, go to Notifications, make sure you have enabled notifications for Outlook.


May I please confirm that you get same behavior after creating new outlook profile? Create an Outlook profile - Microsoft Support please do not delete old profile to prevent the loss of your data.
May I please know the Windows are you using?
Would you mind to kindly provide the screenshot of the outlook version are you using? Kindly go to File-->Office Account
This requested information will help us to narrow down the scenario and find the right troubleshooting.
We are looking forward to your response! Thanks for your precious time. Have a nice day!
Best regards,
Inema | Microsoft Community Moderator