Harder & Steenbeck airbrushes have the standard 1/8inch male bsp thread, if yours has something else, then it must have an adapter on it.
That compressor looks like it has a 1/8inch male bsp output, a regular female to female 1/8inch airbrush hose will connect them up, (the most common type there is- and comes with that compressor you linked)
I would reccomend getting a 1/8inch quick release coupling for the airbrush end, but its not neccesary, just a turn or two of PTFE Tape to ensure the joins are airtight enough.
edit; thats an AS186 compressor - again the most common type there is, I know full well that its a 1/8inch bsp output (universal really apart from badger products)
Honestly I dont understand the problem your having? The airbrush definatly has a male 1/8inch bsp thread, the compressor has the same, you wnat a 1/8inch female connector on each end of the hose and zero adapters. If your hose has a 1/4 inch fitting, then that set of adapters you linked contains the right one to make it fit, but you want one on each end if both ends of the hose are 1/4inch... far simpler to use a hose with appropriate fittings to start with, as more joins is more places to loose air.
Your brush looks like this?
You want this;
And 1-2 turns of this around the male threads before screwing them together;