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if you head shot it yes.
 
Melee / Knock it down / destroy Side Torsos to kill its MechWarrior. : )
 
I've found that later in the game going with an LRM-heavy lance can help with the salvage. Multiple knockdowns, head hits, torso or ammo destruction has allowed me to pretty much batter a pilot to incap status.

You typically do not have the luxury of that much firepower and stability damage at that point in the game, so I add this to the good suggestions made by others here.

Bank morale as much as you can through that mission, until the final fight with the pirate queen. Quickdraw armor isn't that impressive to start with, and being a pirate mech it's pretty squishy for a heavy. One or two good aimed shots is typically enough to take off what you want.
 
Thanks all for the help I thinking that I will get my pilot up to 4 4 3 6 or 4 4 3 7 and then try I know it will take 4 or 5 weeks since I am only able to play about 8 hours a week till school is out
 
For some reason, the Quickdraw usually has a black (untargetable) cockpit for me. The same also applies to Victoria's mech every time.
Such Untargetabke Cockpits are evidence of some very understandable Plot Armor. Victoria is the prime example, if her Cockpit were to be taken out by an AC20 earlier in the Solo-Campaign, it would beggar the believability of her continued appearances in the Restoration Story Arc.