110vac to 24vac transformer question
GND vs ground

GND in DC-electrical is slang for Common or Return voltage -- the reference point they call 0 volts.  It has absolutely nothing to do with the safety earthing on AC mains power, which always uses a green, yellow-green or bare wire.  In AC, current must never ordinarily flow on Safety Ground; it only flows during fault conditions. 

This is just a weird design convention, akin to the way "DC negative" is actually the electron source (what's up with that?) 

So, if you have any background in AC mains, you must have your DC hat, and your AC hat.  


When your AC hat is on, then black is hot and white is neutral, and green/bare is safety ground.  Or you're in Europe and it's brown/blue for hot/neutral.  
When your DC hat is on, then red is positive hot and black is common/return, and that's called GND even though it's not actually.  


Connecting things on the AC mains side to things on the DC side

Never do that.  If there's any connecting to be done, that is the job of the power supply.  So don't do anything mutton-headed like: 


Connecting DC "GND" (black) to AC equipment safety ground (green/bare).
Connecting DC "Common" (black) to AC "common" (white/blue neutral). 


Any of those scenarios will create a hazard situation if a trivial wire failure in the AC mains were to also occur.  For instance a broken/disconnected AC safety ground + a ground fault in an unrelated AC appliance could energize the chassis of your DC device at 120/230V.  We don't want that, and the UL listing for that power supply requires it have internal insulation as an additional safeguard against that happening. Any jumpering between the LV side and the mains side defeats that insulation, voids the UL listing, and thus violates the mains electrical codes. 

GFCI/RCD vs DC systems

You notice that DC systems carry ordinary service current on the thing they call GND.  AC systems do not.  All current is carried on hot(s) and neutral, and only fault current is carried on safety ground.   As a result, GFCI/RCD systems monitor only the hot and neutral wires for equal current, and ignore safety ground altogether.  

We're on the AC mains side here. 


So anything you could possibly do that would cause current to hop from the brown or gray wire, to the green wire, would be extremely bad -- it would disrupt the fundamental systems design of mains power, and also throw a monkey-wrench in the functioning of the GFCI. 

Transformer quality

I recognize the marks plainly: this is a cheap transformer bought mail order, and it's from the Alibaba junkstream.  The CS is a lie; the RoHS is smoke and mirrors.  It doesn't have a UL listing or comparable. It's not legal to install onto the AC mains. (NEC 110.2).

However, listed 24 volt AC transformers of good quality are readily available from any heating and air conditioning supplier. They are universally used for HVAC control and doorbells.  They either mount in a junction box knockout, or replace the lid of the junction box, with the AC mains terminals on the inside and the low voltage terminals on the outside.  Most are only 40 VA, but see what your lights take.