Place the honeydew in a bowl and mash until some of the juice is released; pour into a blender or food processor. Puree until smooth, 10 to 15 seconds. Measure 1 cup of the melon puree, and pour into a jar with a tight-fitting lid. Add vodka, sugar, and green and yellow food coloring. Cover and shake until the sugar is dissolved. Store at room temperature for 1 week.
Strain the melon mixture into a bowl through a paper towel-lined sieve to remove any solid bits. (You may need to do this multiple times.) Pour the vodka into a bottle, and seal until ready to use.