Follow this guide set your chat and email notifications in HubSpot
Overview: Control receiving and disabling your chat and email notifications in HubSpot settings
If you do not have the time to read, here's a video that quickly covers the process
Log in to your HubSpot account.
Click the cog on the top right to go to settings.
On the left side bar under Your Preferences, click Notifications.
Under the Email tab, scroll down to the drop down options and click Chat and email conversations.
Under New unassigned, you have the option to receive or disable both chat and email conversations. Tick the box to receive notifications, or untick the boxes so as not to receive chat and email notifications.