I've owned (note past tense) the Quadra dual A head kit and have a SFII.
As you surmised, the standard SFII's shaft screw-in extension is 7mm dia, and is the one to use with any Eli light. With the Q, as you insert, it will stop itself where the extension/shaft interface contacts the front end of the umbrella tube on it's 'wing'. Given the 'who were they kidding?' nature of the Quadra head's umbrella tube, I'd STRONGLY recommend:
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1. When using the softlighter: So in slowly, go in straight and be careful to STOP the insert process once you hit the end of the extension. It's placement at that juncture is close to flush with the front diffuser anyway. Is it perfect? No, but close.
Unlike the tubes imbedded into the mass other Eli lights, the Quadra tube's thin brittle plastic stand-alone tube won't take any "let see if I can get it in another 0.25"... Try and do that and it will crack. The same with a hurried attempt to mount an umbrella if you start moving the umbrella up/down/laterally as you insert it - that shaft is a nice long torque arm.
2. If using the SFII or any-sized umbrella with the Quadra, I'd strongly recommend using it in combination with a real umbrella swivel to do the work (e.g. find a combination of Manfrotto studs/swivels or a modified Quadra adapter, etc to keep the light along the access of the shaft BUT have the shaft by-pass the Quadra tube completely).
3. Given that the diameter of the flash tube is VERY VERY close to the diameter of the head itself, I'd use the SFII with a reflector or use some other method to keep the SFII sleeve off the flash tube as the latter gets hot.
The Quadra umbrella tube will not tolerate much by way of ham-handedness nor lateral movement of the umbrella before it cracks and once it does, no amount of epoxy will bring a brittle plastic like that back to original working order. Motorcycle fairing plastic-welding maybe..
Now I'm sure 1/2 doz folks will say they use it every day and it's wonderful, but with the Quadra umbrella receptacle designed as it is, it's not a matter of IF it will break in use, but WHEN. Shelling out $500 ish to replace a head for $0.02 worth of plastic (not that the head, minus the app$100ish cost of the tube, is worth more than $25 anyway) would not be fun.
My advice would be that if you intend on keeping the Quadra, want to use it with a SF or large umbrella and aren't concerned about resale, just Dremel-off that idiotic tube and it's 'arm' completely and put together a DIY kit in combo with say a Manfrottro umbrella swivel, etc.