Do I need permission to use another person’s trademark?

Not necessarily. Trademark infringement only occurs when there is a likelihood of confusion. Merely referring to a trademark isn’t infringement. Indeed, you might need to refer to a trademark when describing a good or service.

The doctrine of “nominative fair use” permits this kind of use, so long as the trademark isn’t used more than is necessary. That said, it is a good practice to obtain permission to use third-party trademarks, especially if you’re producing a film for commercial distribution.