Measure the dimensions of your vent cover. Note that the vent cover has sides that will extend through the dropped ceiling panel and flanges that will be pressed against the face.
Don't cut the hole to the size of the flanges, or the vent cover will go right through the hole. Cut the hole to the size of the sides so that they will pass through the hole and the flanges will press against the face of the panel, stabilizing the vent cover and hiding the edges of the hole.
Remove the dropped ceiling panel from the location where you want the vent to go. Cut the desired size hole in the center of the panel with a sharp utility knife.
Install the vent cover in the hole. Secure it to the vent cover using the method provided with the vent cover.
These methods vary among products and may include butterfly clips, screws or spring catches.
Pull the flexible vent pipe out through the hole in the dropped ceiling. If the vent that you are extending doesn't have a flexible vent pipe, buy one at any hardware or home improvement store and attach it to the existing vent.
Connect the flexible vent pipe to the backside of the vent cover that you installed in the dropped ceiling panel. Tape them together using duct tape.
Replace the panel in the dropped ceiling.