Unable to remove a cross-threaded water hose
That's not a quick-disconnect. That is a backflow prevention device (aka vacuum breaker) between the faucet and the hose. It prevents your hose from becoming a water inlet point and introducing non-potable water into the municipal water system. An example of how that would happen is if a firetruck connects to your fire hydrant by your house and turns their water pump on and they suck more water out of the water main than it can supply. That causes a vacuum (negative pressure) in the water line and opening a kitchen faucet will just introduce air into the water main, but a garden hose that doesn't have one of these can introduce mud/fertilizer/pond water/etc. Other things can cause water to backflow, but that's the best example I can think of.

If your garden hose fitting is completely stuck, then I think it would be the next easiest to remove the backflow prevention device and replace it.

That is a set screw, but the set screw is designed to break off after tightening it to prevent a homeowner from inadvertently removing the device which would be against code in many places. The way to remove it is to grind a notch with a dremel tool in the top of the set screw, and then use a flat head screwdriver to back out the set screw.

It is very possible that you damaged the threads of your hose fitting by trying to unscrew this device without removing the set screw. If you remove the set screw and it still doesn't come off, then you'll probably need to replace the entire valve.

But if you're trying to avoid calling a plumber, the final thing I would try is grinding a vertical notch through either the entire backflow device or the entire garden hose fitting and literally split it apart into two pieces off of the threads. It might not work, but if you already have the dremel tool out it's worth a try. They're both made of brass, so it's soft and easy to cut with a hacksaw if that's all you have available too!

3/4" Vacuum Breaker, Brass, FNPT Connection Type