This is pushing my knowledge but trademark infringement would be if you use their logo as your logo. You can’t mislead people into thinking your product comes from them. That doesn’t mean you can’t ever have public domain things in your product.
A trademark (also written trade mark or trade-mark[1]) is a type of intellectual property consisting of a recognizable sign, design, or expression that identifies a product or service from a particular source and distinguishes it from others.[2][3]
I would say an animated intro isn’t a trademark, and even then making a movie with that material isn’t copying someone’s trademark for your trademark. But this could get interesting.