See this forum post and this StackExchange post.
Gmail automatically re-writes the from address with your proper Gmail address before it delivers your e-mail. Forging from addresses is a common spammer tactic and Gmail won’t allow fake from address with OAuth. You can however change the sender name.
Perhaps what you want to do is set the reply-to header. Please see the FAQ:
How can I get my email to show up with a different From: address?
Google supports custom domains with the paid services Google Apps for Work/Government and the free services Google Apps for Education/Non-Profits/Free Edition.
Otherwise, changing the sender address is not possible in OAuth 2.0 mode, and not recommended in Password (Plain, Login or CRAM-MD5) mode. At best, your email provider will re-write the correct email address or give you a connection error. At worst, your IP or entire domain will end up on a Spam blacklist.
Instead, consider setting the reply-to header of the e-mail. This allows the email reply to be automatically addressed to a different email address. Contact Form 7 allows the reply-to header to be set.