How to check a Quick connect control power transformer is working?
Entirely normal.

On the input side, 240VAC (at least in US residential wiring) is center-tapped to yield two 120V legs relative to ground.. These are out of phase to each other, so if you use a phase-to-phase connection you get the full 240 relative to each other but if you use either phase relative to ground it's 120VAC. That's an easy way to make both voltages available for use in the home.

On the output side: Unless there is a reference connection tying one side of the transformer to ground, it is a completely ungrounded, "floating", circuit -- effectively, what you have there is an isolation transformer. Without a ground connection, there is no circuit through ground and hence no voltage, just as you don't read voltage when one of the probes is in mid-air. If you did ground one side, of course, that side would read 0 volts and the other would read as hot relative to ground.

Now you'll ask what the voltage relative to the incoming 220V should be. Without a ground on the output side that too is 0V since there's no circuit. With the output grounded, the result is going to depend on how much phase delay the transformer has introduced -- it's a big inductor, after all.