Google Documents doesn't include in its UI a tool to apply color to the paragraph, cell or table. AFAIK the options available at this time are:
- to set the page background color.
- to insert a drawing
- to insert an image
1. Page background color
This is not suitable due it change the color for the whole document.
2. Insert a drawing
Google Drawings have tools to insert shapes and text objects. It could be used to create text blocks with the required background color. You could create the Drawing directly from the Google Documents UI, just click Insert > Drawing...
. One benefit of this method is that the drawing could be easily edited.

3. Insert an image
You could use an image editor or a editor that allows to save files as images to create the text blocks with the required background color, then insert them to the Google document.