If a character with the shillelagh spell and polearm master feat can they use a d8 for their second attack with a quarterstaff?
For the duration thereafter, you can use your spellcasting ability modifier instead of Strength for the weapon, and its damage die becomes a d8.
When you take the Attack action and attack with only a glaive, halberd, quarterstaff or spear, [...] the weapon's damage die for this attack is a d4, and the attack deals bludgeoning damage.
One could argue that the feat makes it clear that your extra attack with Polearm master says that it deals a d4, period. But one could also say that the d4 is one of the quarterstaff's damage dice, and therefore becomes a d8 because you are making a second attack with the weapon.