Do you need to train birds to go to the bathroom in their cage?

You shouldn't need to, as bird cages typically do not include indoor Plumbing. Also, nature trains birds to do their business without needing human training.

In all seriousness, a cage is a designated area where the bird can 'go' anywhere within it. You will need to clean the cage, though.

As a bird owner, my bird will make a mess in the bottom of his cage (as is expected) and the cage bottom will get cleaned as necessary.