Hi,
Normally on first Run the Messages App asks for an Apple ID for the iMessages account.
At this point if you went to the Messages App > Preferences > Accounts you would see two Accounts.
One is Bonjour and the other is iMessages.
Even if you skipped adding the Apple ID and went to the Preferences to Add other account types, the iMessages and Bonjour ones would still be listed as they cannot be deleted (only turned Off/Deactivated).
An Apple ID can be any Email you have used to register at Apple. In some cases you may have made an Apple Issued ID (using an external email as a recovery email).
Apple issued Email ID include older @mac.com ones, @me.com ones that have been linked to iCloud and @icloud.com ones themselves.
External ones can be from anywhere such as Yahoo, Google Mail, AOL mail and so on.
For some people their email/Apple ID may be with an ISP they no longer use. I once had a deal with an ISP that gave free email accounts back when I used Dial up. I can still access this email accounts to download new emails but I cannot send as I cannot login via Dial-up any more.
However it is my Apple ID and the Password I use at apple for it obviously works.
So...
Your Google ID may well also be your Apple ID.
It could work in the iMessages account AND as a Google Account in Messages.
(i.e. you can add it as both if it is your Apple ID).
IF the Messages App > Preferences > Accounts > iMessages is devoid of an ID you can use your Apple ID.
If these is the Google ID you were talking about it you should use the password you use at Apple with this ID.
If you are adding it as the Google ID the ID must be the same one as it appears when logged in to the web version of Google Mail.
Google Mail uses at least two formats (@gmail.com and @googlemail.com) and these are recognised as two different IDs by the Jabber (GoogleTalk) server involved.
Obviously the Password should be the same as you use at Google.
At this point Google have an issue with Apps that use SSL to login securely.
To allow this you have to login to Google (I do it on the Web Mail page) then access the Account option by using my ID top right as a drop down.
In the Account page you can use Two Step verification - this also does not work well with Messages so if it is ON turn it Off and then Enable the Less Secure Apps option as this will allow you to login using SSL.
Check the two Accounts you have and the ending of the Google ID is use.
(The List can be dragged into any order - except in Yosemite, Bonjour will always go to the bottom when you restarted the App).
Also check the Server Setting tab.
This should list port 5223 and the Server talk.google.com if you come from a older Google Mail ID rather than Hangouts.
9:22 pm Thursday; February 19, 2015
iMac 2.5Ghz i5 2011 (Mavericks 10.9)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
Couple of iPhones and an iPad