In this Google Books search for "a cold yesterday", I found "Johnny had a cold yesterday."
Is this usage correct? Can we use "quality-adjective + adverbial noun" to make "noun group"?
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Yes, but that's not what's being done here.
"Cold" here is used as a noun, referring to an illness where the person has a runny nose, watery eyes, and sneezes. So "Johnny had a cold" means that the person named Johnny had the illness which is called a "cold". "Yesterday" is an adverb saying when he had it: he had it yesterday.
That said, there is a very similar construction that might confuse you. It is common to say things like, "We had a cold winter", meaning that the winter was cold. In this case "winter" is the noun and "cold" is an adjective modifying "winter".
Similarly, someone might say, "We had a very busy Monday": Monday is the noun, busy is an adjective. "Yesterday" can be used as a noun, as in, "Yesterday was my birthday". I've never heard someone say, "We had a busy yesterday", it's not something fluent speakers say, but if I heard someone say that I'd assume they meant "yesterday" as a noun and "busy" as the adjective.