Ovinomancer
No flips for you!
This is on the right path. Look at Stealth skill:
Stealth. Make a Dexterity (Stealth) check when you attempt to conceal yourself from enemies, slink past guards, slip away without being noticed, or sneak up on someone without being seen or heard.
In this case the Golem isn't trying to do anything, it is motionless until its trigger is activated. Its not even trying to be silent, it is silent, not making sounds and not making tracks, so it has the Invisible Condition, impossible to detect by normal means.
If it was an invisible statue no reasonable DM would allow someone to perceive it, you would make them describe what they are doing, search the room, etc. like the invisible object rules. If the golem was completely visible, I might allow a check to tell that is in fact a golem as opposed to an actual statue.
In the case of my example of the invisible stalker, it does make some noise all the time, but with stealth of 10 its passive stealth would be 20, so its mostly silent.
Why do you assume iron golems can be as still as a statue at will? This is a pre-narration. If you insist in this don't roll.