As soon as the character touches another object (which would include drawing the weapon), he would discharge the spell, so it's immaterial.
But, let's assume he had the second weapon drawn and in the off-hand prior to casting the spell...what are you expecting to happen differently? Assuming the touch spell is single-charge only (i.e. shocking grasp, rather than chill touch), as soon as it hits, the spell discharges. If you hit on the first go, you don't need to bother rolling the second attack with the spell, since it's gone. If you miss the first time, you get the second attack. Regardless of what happens with the touch spell, you get your regular routine with the off-hand.
If you touch anything or anyone while holding a charge, even unintentionally, the spell discharges.
So, here's how this plays out:
Step 1: Billy the Bloodrager casts shocking grasp on his first turn and moves into position.
Step 2: On his second turn, Billy uses Quick Draw to draw his battleaxe.
Step 3: Billy's shocking grasp discharges, dealing 5d6 damage to his battleaxe. He rolls a 17, the damage is halved before applying hardness, rounding down to 8. The battleaxe has hardness 5 and 5 hp, so it takes 3 points of electricity damage and gains the broken condition.
Step 4: Billy full-attacks with his battleaxe, suffering from a -2 penalty on attack and damage rolls for his poor life choices.
Step 5: Profit?