The last time I tried doing this (maybe early 2011-ish) I was not able to programmatically send out from Gmail account on GoDaddy. I took code the exact same code that worked on a different hosting company's server and it failed on GoDaddy's server so I'm quite confident that the code was not the issue.
I called their tech support and got conflicting information but I had heard enough to convince myself sending via Gmail was not a for sure supported service on GoDaddy's servers.
I was also given the workaround option of using their web email form which is totally not what the project I was working on could use and underscored the notion that GoDaddy did not fully understand what they talking about.
In the end I found that I was able to just use GoDaddy's SMTP servers and plug in my own "from" address which allowed me to send mail to any email address and to spoof the origination email address. So they end up locking down legitimate email functionality and instead force you to use a highly abusable system instead.
Again, this was a while ago that I ran into this issue but it would be easy enough to check to see if they still allow you to use their SMTP server to send emails.