Install your tile floor all the way up to the vent.
Place the piece of tile you want to cut on top of the vent just like a normal piece of tile. Use a tape measure, pencil and carpenter's square to mark the cutout for that piece of tile.
Spread thinset mortar right up to the edge of the vent on all sides. Set the piece of tile, and let it dry for 72 hours.
Put on safety glasses. Cut along the cutout marks on the tile with an angle grinder that has a stone blade.
Install the tile floor all the way up to the vent.
Mark the tile that you want to cut for the vent just as you would mark it for any other cut, using a tape measure, pencil and carpenter's square.
Put on safety glasses. Place the tile on a tile wet saw's tray, and turn on the saw. Push the tile into the saw's blade, using the marks you made on the tile as your cutting guide. Cut out the waste material.
Dry the back of the tile with an old towel before its installation.