Add everything but the cucumber to a blender or food processor and pulse several times until smooth stopping to scrape the sides as needed.
Slice the cucumber in half lengthwise then scoop out the seeds and discard (or save and toss into a smoothie later). Grate the seeded cucumber. Add the cucumber by picking it up by the handful and squeezing out excess water over the sink, then adding it to the blender or food processor. Pulse a couple of times to incorporate.