In English, when you say “I already said ‘no’”, the word “no” is a direct quotation — you are directly quoting yourself. If you think about it, your exact words were “No” (or actually, “no I don’t” or “know I can’t” or “no, I’m not”…). In Spanish however, we use an indirect quotation, and we need “que” to introduce the indirect quotation (whereas “that” would be optional in English). This will be more obvious with a longer sentence:

A: ¿Me puedes ayudar?

B: No. No puedo ayudarte.

A: ¿Qué dices?

B: (Digo) que no (puedo ayudarte).

Now, isn’t this perfectly logical and much easier than English? Every language has its own way of doing things, so it will be much easier if you stop translating and stop trying to compare everything with English, and simply accept your target language as it is. Why in English do people refer to “a 3-month-old baby”? How is a little baby “old”??? Or why do we say “it’s me” (or “it’s I”) when answering the phone? (For that matter, why can't native speakers agree on whether to say "me" or "I"?) “It” is a neuter pronoun, and you’re not neuter are you? To a non-English speaker, this stuff can seem truly weird. Just about everything is “idiomatic” (not just the colorful "idiomatic expressions" on those long lists that learners are so fond of).

That helps, thank you. I keep falling into the translation trap which I should know better than to do now.

That’s correct “I already said no”, “que” is not use as a question.