Warning: The following article is not good advice. It's sarcasm. If you were looking for reliable instructions about using G+, you've come to the wrong place. I was a little frustrated when I wrote it, for no particular reason, and I was expressing that in sarcasm. Besides that, it was written years ago.
They said it couldn't be done, but I did it. I deleted notifications that I had received from Google plus. (This is not about preventing future notifications. A lot of pages that talk about deleting notifications are actually instructions for turning off future email notifications, which Google+ has settings to do. I'm talking about deleting and permanently removing notifications that you have already received. My method deletes notifications from the Google+ notification button that displays how many unread notifications you have, and from the Google+ notifications page that "View all notifications" shows you.)
Step 1: Back up any of your Google+ content that you want to save.
Step 2: Delete any information Google has about you that may have resulted in receiving the unwanted notification, such as a Gmail contact to a person to whom the notification refers.
Step 3: In Google+ click the gear icon then click "Settings" or use any other way to get to the "Account Settings" page. On that page, find "Disable Google+" and click the "here" link.
Step 4: A page appears with a radio button labeled "Delete Google+ content" to select what you want to delete. Click it. (Trust me. I know what I'm doing, and this is the way to do it. Just don't click the radio button labeled "Delete your entire Google profile".)
Step 5: Click the check box labeled "Required: I understand that deleting this service can't be undone and the data I delete can't be restored." (You want to get this done don't you? It's currently Google policy that there's no way to delete notifications, so you're lucky that there's even this way to do it. Buck up and go ahead.)
Step 6: Click the blue bar labeled "Remove selected services".
Step 7: If desired, enter your reason for leaving as "notifications can't be deleted" or words to that effect, or a complaint about the notification that bothered you and that you couldn't delete it.
Step 8: Click any further confirmation buttons as necessary to complete the deletion.
Now your deletion of Google+ notifications is complete. If you still want to use Google plus, just sign back in, simple as that. (Of course, if you had some Google+ content that you want to use again, such as lists of contacts in circles, you'll have to enter it again. That's why you did the backup in step 1. You did back up your data, right? Well, that was your responsibility, and you were warned.)
[Disclaimer: I have no financial affiliation with Google nor with any other social network service. No warranty of suitability of the instructions provided herein for any purpose is expressed or implied.]