The frog quest is assigned to female characters by drinking from a pool and having a frog talk to you: The frog will give you a golden ball, which doesn't seem to have a purpose.This is a reference to a princess losing her golden ball in a pond in the fairy tale The Frog Prince.
You don't need to have the quest assigned in order to complete it however: In order to complete it, you must talk to a neutral frog when playing a female character. When talking to a frog, the result will either be "Croak, croak, croak" (nothing happens) or else the frog will claim to be a prince and ask for a kiss. If you kiss the frog, you will either grow a wart (-1 Appearance, -1 Charisma) and the frog will turn hostile, or else the frog will turn into a prince (same race as PC).
The reward for completing the 'quest' — i.e. turning a frog into a prince — is a diadem of beauty, a pendant of beauty, a potion of beauty, and approximately 150 gold.
Note that you can talk to the same frog over and over again until you receive the desired result. With enough patience, every frog turns out to have been a prince! Frogs in the swamp work fine for this after they've been pacified with the "Calm Monster" spell or with music. This can be exploited to raise appearance up to 99.
Target: a giant frog (friendly). You talk to the giant frog. The giant frog croaks: "Kiss me and I'll become a prince!" Do you want to kiss the giant frog? [y/N] The giant frog transforms into a handsome human prince! "Ye shall be rewarded for ending my torment!" The prince puts down a bunch of stuff for you. "Now I shall venture back to my home castle and engage in some leisure." (Suddenly his tongue rushes out of his mouth and he catches a fly) "Mmmmhh." "Tasty." With that the prince wanders away.