Yes you can! Look for the ‘active’ class on the template grid item using the inspector in your browser and give that class a background colour in the custom CSS file.
I don’t have a project open with a template grid right now, but for a list view it looks like this:
.mx-listview .mx-listview-item:focus, .mx-listview .mx-listview-item:active
Seems it’s a little bit different for the template grid. But this is the class you’re looking for.