Wash the collard greens and turnip greens thoroughly under cold water. Remove the tough stems from the greens and chop them into bite-sized pieces.
In a large pot or Dutch oven, combine the ham hock, 4 cups of water, and chicken bouillon. Bring the mixture to a boil and then reduce the heat to simmer. Allow the ham hock to simmer in the broth for about 30-45 minutes. This helps infuse the broth with smoky flavor.
In another pot or skillet, heat a bit of oil over medium heat. Add the chopped onions and minced garlic to the pot. Sauté until the onions are translucent and the garlic is fragrant.
Add the chopped collard greens and turnip greens to the pot in batches, allowing them to wilt down slightly before adding more. Stir the greens to combine them with the sautéed onions and garlic.
Strain the broth from the ham hock into the pot with the greens. Add the ham hock to the greens for additional flavor. Stir well to incorporate the broth throughout the greens.
Cover the pot and let the greens simmer for at least 1.5 to 2 hours, or until they are tender. Stir occasionally to make sure the greens are cooking evenly.
Taste the greens and adjust the seasoning if needed. Add more salt and pepper according to your preference.
Once the greens are tender and flavorful, remove the ham hock. Serve the mixed greens hot, with the ham hock on the side or shredded and added back into the greens.