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The short answer is no because by the time Volt Charge resolves, Pyromancer Ascension only one counter on it.

Lets sort this out: You cast Volt Charge from your hand and put it on the stack. Pyromancer Ascension resolves because a sorcery/ instand was cast and it's first ability is also put on the stack, above Volt Charge. Both player's pass and the stack is processed. Pyromancer Ascension's ability resolves and puts a quest counter on Pyromancer Ascension. Then Volt Charge (the proces of casting it is long finished) resolves and you shoot that 3 damage and proliferate to add a second counter to Pyromancer Ascension afterwards.
At no time while casting Volt Charge Pyromancer Ascension has more than 1 counter on it.

Further information from mtg gatherer:
If the creature or player is an illegal target when Volt Charge tries to resolve, it will be countered and none of its effects will happen. You won’t proliferate.

From the official rules:
601.2. To cast a spell is to take it from where it is (usually the hand), put it on the stack, and pay its costs, so that it will eventually resolve and have its effect.