There is a sense in which the moon is symbolic in this excellent short story. For these two middle-aged women who are now widows, and are only able to live the joys of youth vicariously through their daughters, it appears that the moon is a symbol of what they have lost or passed through their age, and perhaps it is also a symbol of the way in which Alida Slade tricked her friend into meeting her husband at night in the Forum.
Note the way in which Alida Slade says, somewhat disparagingly:
"Moonlight--moonlight! What a part it still plays. Do you suppose they're as sentimental as we were?"
Alida here refers to their daughters and the way that moonlight is associated with romance and love. Thus the moon seems to operate as a symbol that recalls their own youth and the role that romance and passion played in their lives, and simultaneously the way that those days have passed for them now and can never be reclaimed. This is why Alida Slade greets references to the moon with such a response.