The conflict depicted in "The Necklace" begins at the opening of the story. Readers are made very aware that Mathilde believes that she should be living a very different life. (Essentially, this is an example of internal conflict given it shows Mathilde's conflict with herself.)
As the conflicts compound, the invitation and no dress, a dress and no jewelry, the loss of the necklace, and the debt for the replacement of the necklace, the conflict extends over the course of much of the text, if not through the end.
Each of these conflicts can be defined differently.
The invitation and no dress qualifies as internal conflict--Mathilde cannot find herself able to go without a new dress.
The dress and no jewelry qualifies as internal conflict--again, Mathilde cannot go to the party because of her own insecurities.
The loss of the necklace qualifies as external conflict-- not only does Mathilde have to replace the necklace, she will go into serious debt to do so.
Essentially, while some may believe that the conflict ends when the debt is paid off, the fact that the necklace was fake introduces another conflict--Mathilde did everything for nothing.