Thanks for all the help. So I was able to retrieve the logs with FileZilla. I tried searching for evidence to see whether the voicemail server attempted to contact the mail server. I used the following search terms: "email", "smtp", "(my email alias)", "(full email address"), "(IP of email server)", as well as many other terms, but no luck.
Unless I am missing something here, it doesn't look like the voicemail server is sending those voicemail messages to the SMTP server.