I think doing an off white paint on the walls, like Sherwin Williams Alabaster, would be beautiful. That way, the quilt would be the focus. An updated look would be to have a solid comforter or duvet and fold the quilt at the end of the bed. Like below pics. Or have the quilt with a solid fluffy comforter folded at the end. Then maybe coordinating pillows in solid colors.
A B&B wall painted one of the paler colors behind the bed would be beautiful, as well.

Thanks Amy Lynn! Great advice, definitely will do the folded quilt and I had not thought about a B&B wall, but love that too!

Oh, sorry Chris! B&B, in this instance, was my short way of saying board and batten. I saw that you mentioned putting in a wall of that behind the bed. I love that idea and was thinking a nice light to med shade of blue that compliments the blue squares in the quilt would be nice.

I'm an avid quilt collector, and I've always thought they look great in rooms with lots of white. White walls, sheers or lace panels or simple shades at the windows, and a white bedspread can really showcase a quilt without competing with it.