Last week I stumbled upon some fittings that are available at Palmer's Pursuit Shop www.palmer-pursuit.com
They have fittings to refill a sodastream tank and fittings to use a paintball tank in the sodastream. I think that both run about $40. There is also an instructional video there that shows how to refill the sodastream tank. Bear in mind though that this is just the fitting, and you'll need a fill station to be able to do it. That will typically run about another $35 and hook directly to your 20lb.
daft is correct that you need a tank with a siphon tube. Yours doesn't have one, so you'll need to turn it upside down, or it will reach equilibrium pressure before fully filling. You need liquid CO2 to get the correct weight into the tank, so trying to fill it with something that hooks to a ball lock QD isn't going to work. Which also brings up the fact that you'll need a scale to be able to know for sure how much CO2 you're getting in there.