Under the default configuration the Zimbra server will accept SMTP connections sending mail to a domain on the server from any computer on the same network, but will deny sending to any domain not on the server. For example:
Zimbra Server: 10.0.0.2/24 domain company.com
Client machine 1: 10.0.0.3/24
Client 1 makes a SMTP connection to the Zimbra server trying to send an email to
admin@company.com from
user@company.com. The Zimbra server will accept this and queue the message.
However anything not addressed to a domain on the Zimbra server will give a "554 5.7.1 <asdf@gmail.com>: Relay access denied" message.
What is likely going on is you have a machine on your network that is infected and just connecting directly to the external email servers. The way to solve this problem is to block all SMTP outbound traffic from any machine that is not your email server.
Where are you seeing these messages going out? Are you finding them in the Zimbra logs, or from bounce back messages you are getting?