"You have new mail" in a terminal is a very old Unix behaviour: the shell (the software which interprets the commands you type in the terminal) regularly checks whether the file
username is, you guess it, the name of the current user) has been modified (precisely, whether its last modification time is ulterior to its last reading time, the mark of a new but as yet unread data).
On a machine which is not an email server, reception of emails that way usually means that the email was generated locally. This is typical of cron jobs: you register some commands for regular execution, and when they get executed, their output is sent to you by email (that's purely local, the data never exited your machine).
See the contents of the file with:
more /var/mail/my_user (in a Terminal window). This will give you some information as to the provenance of such emails. My guess is that they are totally unrelated with your Google story. Note also that I regularly receive (meaning several times per day) a lot of fake emails which purport an alleged security breach, and try to entice me into entering my login and password on some site which may or may not "look like" the genuine site for Google or a bank or whatever. This is common practice among attackers. Your emails might be of the same persuasion: fake emails which do not come at all from Google. Detailed inspection of the header and contents of such an email may reveal the foul play.
If unsure, reset your Gmail password (with a new, strong password); this won't make any harm.