The best place to look for conflict between John and Elizabeth is in the first part of act two, where the two of them are sitting to dinner. The scene starts off nicely enough, even though you can feel the tnesion in the air; both of them are being very careful with their words, hoping to please the other. However, when Elizabeth mentions John going into Salem to mention how he had a conversation with Abby where she told him that none of the girls were actually bewitched, things get really tense.
First of all, the subject of Abby is a tense one to start with. If John goes into town to tell someone that Abby had told him something so important, he is worried that people might question why he was talking to her, and why she revealed that information to him, and no one else. Elizabeth has been pushing him to go for days now, and he is resisting. So, that is conflict number one; it is compounded when she discovers that when Abby told him this, they were alone together, whereas before, she had told him that they were not alone; this touches a sensitive nerve, and makes it look like John has something to hide..
Add onto this the fact that Elizabeth feels that John isn't going into town to tell about Abby because she feels he still has feelings for her. So, they break into a fight about that. She gives evidence that he still cares for her; he tries to deny it, and they argue. When they bring the subject of Abby into the conversation, old wounds are opened. John accuses Elizabeth of being cold and unforgiving. She accuses him of not being truthful.
Lastly, they fight over Mary Warren, and how Elizabeth shouldn't be letting Mary go into town to testify in the court. John wishes Liz would be more stern and authoritative. So, in the opening scenes, we do not see a couple that is peaceful and happy; rather, they are fraught with conflict and tension. I hope that helped; good luck!