Yes and no. I think Tableau means for containers just to be boxes to hold other things, so there is no built in functionality to change the background color as it would just get covered up. That said you should be able to get the same effect by putting something in the container that fills it with the desired color. You could use an image for example, or create a simple worksheet that doesn't actually show anything but has the desired background color. An even simpler solution is to use a text box as you can change the background color of this, and you don't actually need to put text in it.
Does that work? If not a picture might help.