Ezreal's Q can't activate Deathfire touch.

This is intended. While Ezreal's Q is a spell, it's a spell that procs on-hit effects, and because of this it won't proc spell effects. Deathfire Touch is a spell effect, which means that Ezreal's Q won't proc it.

This is the same reason why Rylai's and Luden's don't work with Ezreal's Q.