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“The Green Knight's original challenge was for someone to step forward and strike him with his own axe under the condition that he meet him again one year later to receive a blow in return. Arthur steps up to the challenge, but in order to protect him, Sir Gawain steps up and meets the challenge instead”.
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Whereas both of them are heavily influenced by the art of manga (Japanese comic book drawing), they are not directly related to each other. Hidetaka Miyazaki is notable for his work in the video games field, and is the director of a video game corporation FromSoftware. On the other hand, Hayao Miyazaki is the animator, director, writer, and producer of anime movies - computer-animated movies inspired by manga style. He is the head of Studio Ghibli corporation.