Pass an electronic stud finder, which beeps when it finds denser wood studs, over the wall to find a pair of wall studs hidden behind the drywall to define the edges of the nook. Mark the inner edges of where the studs end with a pencil line on the wall.
Mark with a straightedge the top and bottom of the planned nook opening, connecting these lines with those of the stud edges to create a rectangle.
Cut along the marks with a utility knife, making repeated passes, until you completely go through the drywall; or, score the drywall paper and cut out the drywall with a reciprocating saw. Lift the rectangle away from the wall.
Build a box out of ¾-inch plywood or 1-by-4 dimensional lumber to fit the opening, gluing and nailing the corners and allowing them to dry. Cut backing for the box from ¼-inch plywood or lauan. Hammer the backing to the box with 1-inch brads.
Push the box into the opening and screw it to the studs with brass screws to match your wood tones.
Trim the nook with 1-by-4 dimensional lumber or molding, meeting in a mitered edge or at a 4-inch wood rosette, glued to the wall with construction adhesive and nailed into the stud. Stain or paint the nook box, trim and rosettes.