The Chocolate Frog Card only says that he worked with Flamel on Alchemy. It is assumed that Flamel created the Philosopher's Stone long before he met Dumbledore.
Flamel never revealed the secret of how to make a philosopher's stone, so the study of alchemy is still valid. People are still trying to figure out how. It stands to reason, that Dumbledore tried his hand at figuring out how to make one, and that he did some of his work with Nick. Nick never told him how, and Dumbledore never figured it out completely, but in the end, Nick trusted him enough to share the actual stone itself with him, so he could aid in its hiding and defense.