Gay men who desired to pass as straight would marry a woman and such women were known as beards. Vance is married to an Indian woman but the rumor is he would like to divorce his wife and marry Erica Kirk. So its a double entendre: Vance is a gay man and he will divorce his 'brown' beard and marry one who is white.

Not sure, but I think it is a joke about him abandoning his wife, who is a person of color, to have a fling with this woman, who is white. Could be off.