Can Ezreal's Q crit with Statikk Shiv?

http://na.leagueoflegends.com/en/news/game-updates/patch/patch-42-notes#patch-Statikk-Shiv

Statikk Shiv PASSIVE CRIT :: Passive now only critically strikes when the associated attack critically strikes

Since Ezreal's Q can't crit, Statikk Shiv's passive won't crit.

crit is not an on-hit effect like red buff and sheen. rather, it is an auto attack modifier. you have to actually auto attack to crit.

I believe so, considering GP can apply it with parrrley

No.

Crit isn't an on-hit effect, Mystic shot doesn't critically strike, you need to critically strike to electrocute things with Shiv.

"Parrrley can crit and applies on hit effects as if a melee basic attack."

You'll notice Parrrley's tooltip makes a deliberate distinction between crit and on-hit effects, they are separate things.

Also, Riot patched out all the unintended interactions with things that shouldn't critically strike, critically striking, more than 2 years ago now. Stuff like the bonus damage on Garen's silence or Xin'Zhao's knockup wasn't supposed to crit and used to initially, as did Ezreal's Mystic Shot under rare circumstances.