A Google scan of Fleury's article "Deux problèmes de géométrie de situation" can be found here, and identifies the author as "M. Fleury, chef d'institution", where "M." is the abbreviation for "Monsieur" in French literature of that time. Interestingly, per the introductory sentence, Fleury's article is inspired by the first volume of Édouard Lucas's "Récréations mathématiques", published in Paris in 1882.
Searching for "chef d'institution a Marseille" we find one Henri Fleury as author of the book "Nouvelle théorie des fonctions qui se présentent sous une forme dit indéterterminée", Paris: Noblet et Baudry 1866. The author is described as "chef d'institution a Marseille, licencié ès sciences mathématiques", that is, head of a school in Marseille with a degree in mathematics. It seems likely that Henri Fleury is the same person as the Monsieur Fleury whose article is referenced in the fourth volume of Lucas's book, but we cannot be certain.
The BnF (Bibliothèque nationale de France) has a list of books published by Henri Fleury, which shows that he also published under the names Henry Fleury and Pierre-Henry Fleury, with the BnF treating the last of these as the principal name and the other ones as alternates. The author is described as "professeur de mathématiques", that is, a (high-school) math teacher, with a birth year of 1833. Looking at available scans of some of Fleury's books, most appear to cover high school-level math.
Fleury may have been in correspondence with other mathematicians of his time, as Jeanne Peiffer, "Joseph Liouville (1809-1882): ses contributions à la théorie des fonctions d'une variable complexe", Revue d'histoire des sciences, vol. 36, no. 3-4, 1983, pp. 209-248, mentions a "lettre du 16 janvier 1866 à M. Fleury de Marseille". Again, we cannot be certain that this pertains to the same person.