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"?
Cold yesterday is not a noun phrase in the examples in the link. Cold refers to an illness, and yesterday refers to the time he had the illness and is not part of the noun phrase. The sentence is correct and can also be written as Yesterday, Johnny had a cold.
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.