I recently discovered that I have Lilies of the Valley plants that sprung up around the edges of my island. Am I able to combine four of these lilies in a diamond formation to breed more lilies plants?
No, lillies of the valley (not to be mistaken with normal lillies) cannot be bred or cloned. There is only one variety (white) which has a chance to appear each day your island is a 5 star island, according to Isabelle's island eval. If your star ranking drops back below 5 stars, it will stop having a chance to appear daily on your island's cliffs. Many sources support this, including:
...it cannot be bought as a seed, nor can it be bred using hybrid flowers. Instead, this flower only appears and blooms once you have achieved a perfect 5 star island eval rating for your town.
Nookipedia also confirms that breeding them together to create more does not work, as they do not breed with other nearby lillies of the valley:
The plant does not breed when placed next to other lilies of the valley, but increases the chance for hybrid flowers if placed nearby other plants.