If you have the space, why not set up a mail server of your own? There are a number of limitations with using GMail that make it an unattractive proposition I suggest. That aside, when you set up your Moodle did you specify an SMTP server? Likely not, so have a look at this document too. You may want to pay attention to the permissions and capabilities settings as well as the items you want to send notifications about as well.
General help
SMTP Settings for Email
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