If a Contact Form isn't sending emails properly, setting the Mail Settings to use Simple Mail Transfer Protocol (SMTP) may resolve the issue. By default, Joomla 2.5 and Joomla 3.0 uses the PHP Mail() function to send emails from the server. The settings can be modified via the Joomla admin area.
The key point is using the proper SSL port number and using the proper Google account information including the username and password. In the top menu, hover over Site and then click Global Configuration, go to the Server tab. Locate the Mail Settings to configure your SMTP settings. Here you will enter your Google Apps Domain E-Mail account information.
Google Apps SMTP Email Settings
Mailer → SMTP
From Name → Your site name or your name Administrator of This Site
Sendmail Path → This should already be present if not it needs to be /usr/sbin/sendmail
SMTP Authentication → Select Yes
SMTP Security → SSL
SMTP Port → 465 (If SSL port 465 does not work for you try and use port 587 instead)
SMTP Password → Your email password
SMTP Host → smtp.gmail.com
After adjusting the settings click "Save and Close". Now you can go to contact form and send email. Be sure to test your Joomla site and ensure emails are sending properly.