The answer to this one can be found at the very, very end of the story, and it is only mentioned briefly, so you have to look closely as you read. We don't learn much about Alcee's current life until the end of the story itself. Before then, all we knew was that he was an old boyfriend of Calixta's, who still harbored a flame for her that was rekindled upon his visit to her home. It isn't until sections IV and V that we learn anything about his own personal situation. He is married, and has a wife that is named Clarisse, who is away visiting Biloxi for "health and pleasure." He does have children, but here's the thing--their names aren't listed. The text only mentions this:
"if [Clarisse] and the babies liked it at Biloxi, to stay a month longer...she and the babies were doing well."
We don't get a name, only a mention of him having "babies." This implies that either they have more than one child that are very, very young, so probably about a year apart, and who are still babies, OR, that they have twins that are still babies. But, there is no mention of a child's name. Calixta's child is named Bibi, but that is the only child name given.
I hope that helped; good luck!