The Crucible

Are they likely to find a resolution to their conflict? Why or why not?

I assume you are talking about their argument at the beginning of Act I scene 2.  I do not believe that at this point they are trying to find a resolution to their conflict, they are rehashing arguments that have been repeated again and again.  John feels guilty and Elizabeth is not ready to forgive or admit that she may have any fault in their marital problems.  John feels that he is always doubted in everything he says and that he must walk "on tiptoe" around the house constantly.  And Elizabeth feels she cannot trust him accusing him of lying to her about seeing Abigail in a crowd, when he spoke to her alone.  They will not reach any resolution until both are ready to admit their faults and look for a solution.  This comes later for Elizabeth after first her and then John's imprisonment.  John actually has to go the point of death to get to a place where he can feel that he is a "good" man again.

I agree with Jilllessa...Puritan society did not leave much room for forgiveness and moving on after making a mistake.  They are rehashing old arguments, and it is not until they are both in jail that they make peace with each other over Abigail and move forward in their relationship which does become much deeper and stronger in love and respect.  Ironically, John will die and Elizabeth will only be saved from death because she is carrying their last child.