Place the water and sugar in a 1 quart saucepan. Add the chopped cucumber and jalapeños, and bring to a boil. Lower to medium flame and simmer 2-3 minutes.
Remove from heat and allow to cool completely. Strain out the cucumber and jalapeños, reserving the syrup and discarding the vegetables.
Fill an 8 ounce glass with ice cubes. Squeeze 1/2 of lime’s juice into glass. Pour 1/4 of syrup into glass and then fill with club soda or sparkling water.
Stir the with a tall spoon and garnish with jalapeños and cucumber.