The play does not entirely answer this question clearly. Oberon and Titania accuse each other of having been involved with the opposite-sex member of the other couple, but neither will admit to such an involvement:
Why art thou here,
Come from the farthest steep of India,
But that, forsooth, the bouncing Amazon,
Your buskined mistress and your warrior love,
To Theseus must be wedded, and you come
To give their bed joy and prosperity?
How canst thou thus for shame, Titania,
Glance at my credit with Hippolyta,
Knowing I know thy love to Theseus?
Didst not thou lead him through the glimmering
From Perigouna, whom he ravishèd
And make him with fair Aegles break his faith,
With Ariadne and Antiopa?
These are the forgeries of jealousy
Titania first claims that Hippolyta is Oberon's mistress, or at least that she had been before she got together with Theseus. Oberon, she says, has come to Athens in order to bless the marriage of Theseus and Hippolyta, presumably because he wants his former mistress to be happy.
Oberon gets angry and calls Titania a hypocrite but without quite admitting exactly what happened with Hippolyta. He talks about his "credit" with Hippolyta, which means his reputation concerning what happened with her, not what actually took place. He tells Titania that she repeatedly seduced Theseus away from the other women he had been with. (In Greek mythology, Theseus is especially famous for having courted many women, leaving each for another as he grows bored or finds it to his advantage.) Titania denies this accusation, saying that Oberon is imagining things because he is jealous: "these are the very forgeries of jealousy." Like forging a signature, a forgery is a lie, so Oberon's jealousy, she says, is lying to him about what happened. Both of them, then, accuse the other, but both of them either deny or refuse to admit that they actually did get involved with members of the royal couple.
Considering how angry Titania and Oberon get at each other, it seems probable that neither is completely making things up. However, the final interpretation of what happened is up to the director or critic, since the play is not definitive on the matter.