TextBlock derives from FrameworkElement and there is no background property for FrameworkElement. So there is no background property for the Textblock class.
Generally, we will provide the background effect with an additional element looks like a container for the TextBlock. For example, we could use Border or Grid like @macintoshpro mentioned in his reply.
If you are using border, it looks like this:
<Border Background="Red"> <TextBlock/> </Border>
Sure, glad to hear that it meets your requirements. If this helps you, please accept it as answer. It might help other people with the same problem.