Though demur and demure look similar and are pronounced similarly (sometimes the same), each have a completely different meaning.
Demur (verb) means: to object, or, take exception. | “Aren’t you going to try some fruitcake?” He demurred, recoiling from his dish with disgust, “You expect me to eat this? What are you trying to do, kill me? Why, I’ve half-a-mind it’d bite me back!”
Demure (adj.) means: reserved, modest (in a affected way), or, coy. | After the grisly duel concluded, the victor sheathed his blade and gave a demure bow to the crowd.