From the description of the Helm of Teleportation:
"you can use an action and expend 1 charge to cast the teleport spell from it"
So, that establishes we are in fact casting a spell. It is not "innate". It is not "spell-like effect". We are casting a spell, but we are not consuming a spell slot.
Now, that spell in particular, Teleport, has only a Verbal component.
So, if a Sorcerer (or Metamagic Adept) with Subtle Spell expends a Sorcery Point, they can use their attuned Helm of Teleportation in a subtle manner, then they will simply POOF with no forewarning whatsoever, and thus, no chance of Counterpsell. Agreed?
I'm 90% sure this is correct but would like confirmation, and if it is not correct, where is my logic or interpretation of the rules flawed?
Now, what if it was a Scroll with Teleport instead? This I'm less sure of. Clearly others can see they are reading from a scroll. But, I could see one ruling that the verbal reading of the scroll is silent, and by the time you see something that can be Countered, it's taken effect, and they've POOFed. So again, no Counterspell option.
However, I can see my players getting super cheesed over that...but unless I'm missing something RAW these would be treated the same?
Asking for a BBG... ;-)