The poster is correct.
This type of reframing of an adjective as a noun used in the English-speaking world, and is beloved by advertizers:
A mouthful of "awesome" in every bite!
Auditioner: Okay, can you portray the same character, but make him ... sad?
Actor: Sorry, no, I don't do "sad". Unless it's "cheesy sad"; I'm a comedian!
Auditioner (to others): Did you hear that? We have a clown here (which I mean literally as well as pejoratively) who says he doesn't do "sad", unless it is "cheesy sad".
A: I went on a date with a really intelligent guy, but he intellectualized every topic we talked about.
B: I hear you. Intelligent is good. Nerdy: not so much.
We probably accept this kind of thing by imagining the word to be completely outside of the syntax, as if wrapped in quotes (which is why I wrote all the examples that way).
The ability to use pieces of language as nouns is necessary, because it allows us to talk about language. We can say things like:
B: What on Earth is "blurchmoop?"
"up on" is a different preposition from "upon".
We cannot stop ourselves from saying sentences like these because we don't know the lexical category of blurchmoop, or because "up on" isn't a noun, and so cannot serve as the subject to the verb "is".
A piece of quoted language can serve as a noun, or even a verb, to make these kinds of sentences work, and as a byproduct, it lets us say things like "awesome lives here" or "I'm with stupid".
Also note that "amazing" can in fact be regarded as simply the gerund form of "to amaze". Thus, "where amazing happens" can be interpreted similarly to "where cooking happens". However, this is not the interpretation which jumps out at me; and if I wanted to convey that meaning, I would say, "where amazement happens". Because "amazing" is a common adjective, we do not use the gerund form "amazing" in contexts where it is not clear that it cannot possibly be the adjective, such as, "He goes around amazing everyone with his skill". So if "where amazing happens" is used by someone with the intent of invoking the gerund meaning, patterned after "where cooking happens", that someone must not be a native speaker of English. By dumb luck, however, that someone has created a glib phrase suitable for an advertizement.
Another thing: "the amazing" is definitely a noun.
Every morning, Bob the Contortionist does the amazing; he bends over backwards and bites his own left ankle.
The slogan in the poster could be expressed as:
Where The Amazing Happens
Finally, check this out. There are some adjectives in English that serve as nouns also, such as various -ible and -able words: convertible, deductible, dirigible, ...
Something can be convertible, and we can have "a convertible" and "the convertible".
Now imagine if such a word were used for something uncountable. For instance, gases are like liquids, but they are compressible. What if we coined the word "compressible" as a way to say "gas"?
Then we could quite perfectly say something like this, without any article "the" or "a":
This pipe is where compressible escapes from the tank.
Which is not so different from "where amazing happens". So maybe "where amazing happens" is not that far fetched; perhaps we do not have to resort to hypotheses about quoted material being treated as a noun.