Try this...

Use smtp.gmail.com for the server

Change port to 465

Require secure connections checked

Basic authentication with your gmail username and password (make sure you use the full email address for username with @gmail.com at the end)

If that doesn't work, you could try installing IIS locally on your machine and run an SMTP server. Allow relay only for localhost and you should be able to use that for the server in database mail without an issue. Just make sure there isn't a firewall blocking the traffic outbound. It's an easy thing to try and it will probably work.