To the jar of a food processor, add the chopped cucumber and pulse until the mixture turns smooth. Run the puree through a sieve, discarding the pulp, and reserving the cucumber juice. You should end up with 1 1/2 cup of cucumber juice; set aside.
In a small saucepan, combine the sugar and water. Heat on medium low until the sugar dissolves.
To a pitcher, mix together the cucumber juice, tequila, triple sec, lime juice and jalapeño . Add half of the simple syrup and do a taste test. Adjust according to taste: add more simple syrup if you'd like it to be a bit sweeter.
Transfer to the refrigerator so the jalapeño has time to spice the mixture. Check the mixture every 15-20 minutes and do a taste test. I was happy with the spice level at the 1 hour mark.
On a plate, mix together the salt, sugar and lime zest. Run a lime around the edge of the glasses to moisten; dip the rims in the salt and sugar mixture. Add a few ice cubes to each glass and divide the margarita mixture between the glasses. Garnish with a few slices of jalapeño or lime or even a few slices of cucumber.