Measure the distance from the floor to the center of the vent. Mark the distance on a piece sheetrock and draw a horizontal line across the board at the center mark.
Measure the distance from the adjacent walls to the center of the vent. Mark the distances on your sheetrock and draw a vertical line across the board at the center mark.
Set your drywall circle cutter to the diameter of the vent. Embed the center point of your drywall circle cutter where your vertical and horizontal lines meet, and rotate the cutter to cut a hole.
Cut the sheetrock in two at the center of the circle with your utility knife -- this will leave you with a semicircle in each piece.
Place the lower piece against the wall so that the lower half of the vent fits in the semicircle. Attach the board to the wall studs by driving drywall screws every 8 inches.
Place the upper piece against the wall so that the upper half of the vent fits in the semicircle. Attach the board to the wall studs by driving drywall screws every 8 inches to completely encircle the vent.