• As a 12th-level monk, I get 1d8 unarmed damage dice.
• I have a quarterstaff as my weapon.
• I have PAM feat.

Should my Bonus Action attack with the staff (not unarmed strike), do 1d4 damage (PAM) or 1d8 damage (monk weapon)?

I would say that the intention is "No, the Polearm Master feature is not affected by the Martial Arts Die."

I base this on this Sage Advice  ruling on whether the extra damage of the Shillelagh affects the Polearm Master butt strike.

If I cast shillelagh on my quarterstaff and have the Polearm Master feat, does the bonus attack use a d4 or a d8 for damage?

The benefit from Polearm Master applies to the opposite end of the weapon and always uses a d4 for damage rather than the weapon’s normal damage die. This is true for a quarterstaff enhanced with shillelagh just as it is for a normal one.

If actual magic can't affect it' I'd say Martial Arts die wouldn't affect it either.

That said....There is almost zero game mechanical difference for allowing it on a standard 2014 Monk built around Dexterity;  The requirements to use Polearm Master's Bonus Action Strike are more restrictive than to use the 2014 Monk's Martial Arts / Bonus Action Unarmed Strike,. The latter can be used in any situation where you'd qualify to use the Polearm Master Bonus Action feature.

Edit: A quick survey of Monk/Subclass features I can find that would be usable with the Martial Arts Bonus Action Unarmed Strike but not with the Polearm Master Bonus Action Strike:

• Monk 6 / Ki-empowered Strikes
• Way of Mercy 3 / Hand of Harm
• Way of the Ascendant Dragon 3 / Draconic Disciple / Draconic Strike
• Way of the Astral Self 3 / Arms of the Astral Self
• Way of the 4 Elements 3 / Elemental Discipline / Fangs of the Fire Snake
• Way of the Open Hand 17 / Quivering Palm
• Way of the Cobalt Soul 6 / Extort Truth
• Way of the Cobalt Soul 17 / Debilitating Barrage

In any other case than "I really don't want to touch this creature I'm fighting with my actual body parts"  It's less useful to use the Polearm Master's Bonus Action Strike over just Monk-punching it. So just handwave and call on "Flavor is free."