Yes, it is the theme from Rio Bravo, but that particular piece is not the actual Deguello bugle call that the folks at the Alamo heard.
Here’s the bugle call from ‘The Alamo’
https://www.youtube.com/watch?v=QKD5TgHWyxM
Just the bugle call:
https://www.youtube.com/watch?v=wKm0UoXZWmc