Preheat the oven to 325F. Lightly grease one 9-inch round cake pan and set it aside.
Slice the cucumber in half and remove the seeds with a spoon. Reserve 3-5 slices for garnish, then cut into large chunks then puree in a blender until smooth. Add the lemon zest and juice and mix well.
In a large bowl, cream the butter and sugar with an electric mixer for 1-2 minutes until light and fluffy. Beat in the eggs, one at a time, then beat in the vanilla.
In a separate bowl, mix together the flour, baking powder and salt. Stir 1/3 of the flour mixture into the egg mixture, then mix in 1/3 of the cucumber. Repeat until everything is combined and blended together.
Pour the batter into the prepared pan and bake for 30-35 minutes, until a skewer inserted in the middle comes out clean.
Leave to cool in the pans for 5 minutes, then turn out onto a wire rack to cool completely.