Can you move a bluebird house with baby bluebirds in it to keep away from cats?

No, baby birds should never be moved from their nest. Birds have an excellent sense of smell and relocating a nest can cause the adults to abandon it. Additionally, moving a nest puts babies at risk of injury or death. If you are concerned for the baby bluebirds' well-being, the best thing you can do is make sure they have plenty of food and water nearby and keep an eye on them until they can leave the nest on their own.