There are several conflicts in the story. One of the main conflicts is the ongoing struggle that Leo has within himself in regards to his own nature. The process of matchmaking has left him feeling unsettled, and his conversation with Lily reveals some things about himself that upsets him. He struggles with two main things in the story: his inner nature and whether he truly is capable of love-either of God or a woman, and his slight disgust at the commercial nature of the matchmaking process that he has undertaken. Leo struggles with both of these throughout the story, the second one especially as he suspects Salzman having manipulated his infatuation with Stella.
Other conflicts in the story exist between Salzman and Leo; Leo is uncomfortable with how businesslike Salzman peddles matches, then is very upset that he misrepresents Leo to Lily. Then there is conflict between the two in regards to Stella; Leo wants to meet her, Salzman hesitates, and they have conflict there until Salzman acquiesces.
Those are some ideas of the basic conflicts that exist within the story; I hope that it helps!