Stop your quilting on the blocks and the borders at least 1/2 inch from the seam. You'll sew the front seams together like normal but on the back you'll hide the seam with a folded sashing strip. It helps if you trim your batting away from the seam lines 1/4" also but be very careful not to cut your top or backing while trimming.
You could also butt the borders up to the blocks and zig zag the borders on but them you'll have to hide the front & back seams with sashing.