Tips on how to install laminate over existing flooring
I would cut a 3/4" strip of flooring and use some construction adhesive to fasten it to the edge of the stair and install the stair nose up against it with some adhesive. That won't expand and contract like the flooring might. Then fit the perpendicular pieces snug up against the back edge of the nose or underneath it if it's designed to overlap a bit. You might need a drop or two of adhesive on a few of those members to hold them down at the edge of the nose. Good luck.