For the hand-quilting steps, use a quilting needle and quilting thread, which slides easily in and out of the fabric and is less prone to tangles.
Another option is to use regular thread and pull it through beeswax, commonly available at sewing-supplies stores. (Clover makes a Hawaiian Quilting Needle Set with needles for applique, basting, and quilting; available at clover-usa.com.)
Baste 1/4 inch inside motif's perimeter with a 1/2-inch running stitch. Remove pins, and smooth the motif as you go.
Applique motif: Using an applique needle, turn 1 edge of motif under 1/8 inch, and secure with applique stitches (bring needle up through base fabric and motif at its folded edge, then down through base fabric only). Repeat, turning under and applique-stitching entire motif. (Note: In Hawaiian quilting, applique edges are not turned under before they're applied. To applique curved areas, turn under less fabric, and work the stitches closer together; you don't need to snip the curves.)
Remove basting stitches. Press motif and base fabric with an iron.
Cut a 24-inch square of wool batting. Lay the second blue fabric square on a flat surface, and place batting on top. Then lay appliqued square face-up on batting.
Baste through all 3 layers along the horizontal and vertical axes with a 1/2-inch running stitch. Then baste in parallel lines, about 6 inches apart, to make a grid. Put fabric in a quilting hoop.
Quilt around motif's perimeter: Thread a quilting needle with blue quilting thread, knotting at 1 end. Starting near the center of motif at an edge, bring needle up from the bottom, and quilt around motif's perimeter, closely following its contours. Complete the area within the hoop, and then shift hoop to work around the entire perimeter.
Quilt echo lines around motif: About 1/2 inch outside the first quilted line, quilt a second line echoing the first. Continue quilting a third echo line a 1/2 inch outside the second, and so on, stopping about 1 inch from the edges, which will be covered by a border. (You can also start 3 needles at each of these 1/2-inch intervals, as shown, to have the echo lines progressing simultaneously to help save time.) Tie off knots as needed at the back. When quilting is complete, remove basting stitches.
Quilt interior echo lines: Using green quilting thread, follow the procedure directly above to quilt echo lines at 1/2-inch intervals inside motif as well as to delineate the flower contours in motif's corners. Trim 1/2 inch off all sides of quilted fabric square so that it is 23 inches square.