An interpersonal conflict is not just a disagreement between two people or two groups, but it is specifically a disagreement "involving significant resentment and discontent" (Bell, "Interpersonal Conflict Definition").
One example of an interpersonal conflict can be found in the very first scene of A Midsummer Night's Dream. Hermia and her father certainly disagree with each other and that disagreement is causing a great deal of "resentment and discontent." Hermia's father Egeus wants Hermia to marry Demetrius and has given his consent, but Hermia wants to marry Lysander. More importantly, Egeus's preference for Demetrius is completely arbitrary as there is no real reason to consider Demetrius to be a better match for his daughter than Lysander. As Lysander points out, he and Demetrius are complete equals in terms of both social status and wealth; Lysander may even be wealthier than Demetrius, as we see in Lysander's lines:
I am, my Lord, as well derived as he,
As well possess'd, my love is more than his;
My fortunes every way as fairly rank'd,
If not with vantage, as Demetrius'. (I.i.101-04)
Regardless of the arbitrariness and irrationality behind Egeus's preference, he still continues to see things his way rather than his daughter's. He even threatens her with either banishment to a nunnery or execution should she continue to disobey his decision. Since Egeus and Hermia not only continue to disagree throughout the play but the disagreement also causes both "resentment and discontent," this is clearly an excellent example of an interpersonal conflict.
There are many interpersonal conflicts at play in "A Midsummer Night's Dream." There are three sets of human couples, as well as the fairy couple, Oberon and Titania, that could all be used to discuss interpersonal relationships in the play. First, Theseus and Hippolyta are to be married within a few days at the opening of the play. But they hardly know each other well because Theseus conquered Hippolyta's land and then took her up as his bride. Second is Lysander and Hermia who have the law against their desire to be married, which strikes a bad note between Hermia and her father. Then, Helena was once Demetrius's girlfriend, but he changed his mind in order to marry Hermia. Not only does this throw a wrench in Helena's and Demetrius's relationship, but between Helena's friendship with Hermia. Finally, the fairy monarchs Oberon and Titania are fighting over custody of a young boy which wreaks havoc on the fairy world. Summarily, there are conflicts in almost every interpersonal relationship there is: enemy/enemy, father/daughter, friend/friend, boyfriend/girlfriend, and husband/wife.