The Silke Dilemma is not a true quest, but it is a necessary prerequisite encounter towards being able to invite the bard Garrick into your party.
Speak with Garrick outside the Burning Wizard inn in Beregost, and he'll explain how his mistress Silke is being threatened and needs defending. Apparently she was supposed to perform at Feldepost's Inn but didn't show up and as a result is being hunted by Feldepost's thugs.
Garrick offers 300 gold pieces for your help. If you agree he'll take you to Silke, who is outside the Red Sheaf Inn. Once there, he introduces her. When you speak with her she raises the reward to 400 gp and warns you not to give the thugs a chance to explain themselves.
The 'thugs' arrive, acting as if Silke had arranged to meet them. They claim they are here to deliver some gems to Silke. Silke interrupts them and demands that you kill them.
The dilemma is choosing whom to believe.
If you read the interactions it should become quite clear that Silke is the villain in this situation, while the 'thugs' (Glayde, Faltis and Tessilan) are both innocent and bewildered. If you refuse to kill the them, as she demands, Silke turns on your party, which turns out to be more profitable anyway as you collect the promised 400 gp from her corpse along with other loot, including a quarter staff +1, and avoid the -2 penalty to Reputation for having defended the guilty from the innocent.
Speak to any one of Glayde, Faltis or Tessilan to receive their thanks and a potion of defense.
Once Silke is dead, Garrick comes to his senses and, if you speak to him, asks to join your party. He will remain standing outside the Red Sheaf Inn indefinitely (for several in-game months, at least) until you speak to him.