In "Desiree's Baby," is Valmonde the name of the characters or the name of the place?
It's both. At the beginning of "Desiree's Baby," the narrator states that Madame Valmonde is on her way over to see the baby. In the second paragraph, when Chopin introduces Desiree's mysterious background, she writes that Monsieur (Mr. Valmonde), when
"riding through the gateway of Valmonde had found [Desiree] lying asleep in the shadow of the big stone pillar."
During the story's setting, it was not uncommon for upper class Americans (especially in the plantation South) to name their family estates after themselves or their ancestors. In Armand's case, his last name is Aubigny, but his plantation is referred to as L'Abri.