When in the Exceptions module, there is a column called "Created" that tells you the date and timestamp for when that particular exception was generated.
To access this, go to the Exceptions module:
Then, when in List view for exceptions, make sure the "Created" column is enabled by clicking the columns icon:
Then, make sure the "Created" column is selected:
Hit Save, and now this column will show up in your exception results:
You can see the timestamp for the date by hovering your mouse cursor over the date, where a pop-up box will appear with the date, time (PST), and time difference from UTC.
Export reports and scheduled alerts will include the Created column, if it is enabled in the UI.