Thank you so much for your reply, Ralph
Are you using 2 Step Verification ?
Yes, I had.
I disable and tried login in iMessage(OS X).
DIDN'T WORK
DO you have App Specific Passwords ?
No, The moment that started the problem hadn't a App Specific Passwords.
But for test, I created one App Specific Passwords
and tried login in iMessage(OS X) with both passwords.
DIDN'T WORK
Whilst iMessages must use and iCloud Valid ID it is not part of the iCloud items (see System Preferences > iCloud as there is nothing here for Messages).
Unfortunately this means that signing in and out of iCloud here does nothing for the iMessages account.
My normal password (AppleID) always worked before. I didn't change anything and iMessage (OS X ) don't work (in my other devices, works fine.
Try a different Apple ID in iMessages if you have one.
I create now a new Apple ID and tried login in iMessage(OS X).
DIDN'T WORK
In the Apple Icon Menu top left use the About this Mac item.
Can you see the Serial Number on the Overview tab ? (we don't need to know or see it, just that is is there).
Yes, I see the Serial Number
Try iMessages when in a SAFE BOOT
I tried this yesterday, and tried login in iMessage(OS X).
DIDN'T WORK
Try both the different ID and your regular Apple ID ins a test Mac User Account (See system Preferences > Users and Groups and enable the Guest account).
I tried with Guest account.
Works with Apple ID (2 Step Verification DISABLE)
Works with Apple ID (2 Step Verification ENABLE, not in first time but after a few attempts I got a login )
For this reason, this is a problem with authentication caches in my OSX account.
But from here I don't know what to do anymore.
I already had a problem similar to iCal. My gmail account was not accepted in ical, (it worked well for email). The solution here was clean Installed OSX, but at this point I can not do that.
Thank you for your help!