Add the cucumber to a blender and blend on high until pureed, about 1 minute. (If your cucumber doesn't blend easily, you can add up to 2 tablespoons of water to help it along).
Pour the cucumber mixture into a strainer over a bowl and press the juice through, leaving the solids behind. (You should have about 1/2 cup juice.)
In a pitcher, stir together the cucumber juice, vodka, lime juice and agave. Pour in the sparkling water and stir. Fill 4 rocks glasses with ice and divide the spritz among them. Garnish each with a cucumber spear and mint sprig and serve immediately.