Cut one inch off the end of the carnation stem and immediately place the stem in the green water. In time, the flower will drink the water, carrying the food coloring up the stem and into the petals.
For St. Patrick's Day, you can arrange white carnations with green ones in a pretty vase, or you can pin a green carnation to your shirt or jacket.
From Shamrocks, Harps, and Shillelaghsby Edna Barth
Tips from How to Dye Carnations for ANY occasion
by Mary Rees
You can also dye a carnation 2 different colors. Split the stem in half and put one half in one color of water and the other half in the other color. You will get a double colored flower.
Make sure they are fresh flowers that are not fully opened yet. You may leave the flowers out or place them in the refrigerator overnight. Usually the warmer temperature will cause the flowers to drink the water faster. Depending on the air and water temperature you will visibly be able to see the color going up the stem and into the flower in as little as an hour.
Submerge 2-3 inches of the stem bottoms into a large bowl of warm water and cut at least 1/2 inch of the stem off using sharp shears. Cutting the stems off under water re-opens the stems which helps start the flower sucking water back up the stems. This also dislodges any air bubbles in the stem which can kill a flower.
Note: Using the same technique, you can dye celery stalks all sorts of interesting colors.