I get an App Store notification on my MacBook literally every time I use it. The sound makes my dogs go crazy, because it's the same noise as the camera on the front door bell. Of course in true Apple fashion, there is no "App Store" notification setting on "System Preferences". The only "solution" I can find is to allegedly disable notifications until some future date. E.g. open Terminal and enter something like this:
defaults write com.apple.appstored LastUpdateNotification -date "2099-12-12 12:00:00 +0000"