Hi,
The Messages App can use your Apple ID that is linked to iCloud to connect via the iMessages service.
It can also use Valid AIM Screen Names or Jabber ID and Yahoo ID.
Google run a Jabber server so you can add that as a separate account in Messages (or allow it from System Preferences > Internet Accounts).
Messages uses SSL to login to most Jabber servers on port 5223
Google however have decided the SSL is not secure enough and created their own Aouth Tokens.
However that need 2 Step Verification to work and that cannot be used if you need to use "Less Secure Apps"
As it is a message/alert from Google it is unlikely that the Google ID you are using is being used as the iMessage/Apple ID and it is for a separate Google Account.
The way you have posted does not make it clear if you are using a separate account or it is the Apple ID used in the iMessages account.
9:30 pm Thursday; April 21, 2016
iMac 2.5Ghz i5 2011 (El Capitan)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
iPhone and an iPad (2)