This is an interesting detail that helps contribute to local color of the novel. First of all, a quadroon is a woman who is one-quarter black and three-quarters white. What would be significant about this is that A) it shows the status and wealth of the Pontellier family in the sheer fact that they can afford to have full-time help with the children and B) it illustrates the status of blacks in Creole New Orleans at the turn of the century. The fact that she only a quarter black would have given her a bit more status because she would look more like the powerful whites, but they could always hold her race against her as a means to wield all of the economic power. In the South, no matter how light a person was, if they had any black ancestry they were considered blacks and were subject to the unfair laws, standards, and practices of the whites.
It gives another feel to the setting as a woman with black ancestry has a profession as a servant.