The episode is Family Guy Viewer Mail 1 from Season Three
That episode contains 3 different segments, and the second segment is called "Super Griffins", where the Griffins develop superhuman powers after being exposed to toxic waste from a tanker truck.
Peter Griffin has the ability to shapeshift (in that segment he transformed into a Tyrannosaurus Rex).
Lois Griffin becomes super strong, having superhuman strength.
Stewie Griffin's cranium doubles in size, having telekinesis.
Brian Griffin has the ability to move at superhuman speeds.
Chris Griffin has the ability to start fires by thought, developing pyrokinesis.
Meg Griffin has the ability to grow her fingernails longer or shorter on command.
iDate a Bad Boy
Meg.
Only in fiction and art. There never has been real griffins.
Jasper
04x18 - The Father, The Son, and The Holy Fonz
They were exposed to toxic waste.
Boys Do Cry
iDate a Bad Boy
Mother Tucker
From Method to Madness
Bette Midler | The Rose
Meg.
Brian griffins house of pain
Dog Gone (US season 8/UK season 9)
Griffins are never referred to in the Bible .
Spikes of gold
Baby griffins are called chicks.