Does this sounds natural ”Their performance is quite a neck and neck”

No, this isn't the right way to use this expression. First of all it isn't a noun, it's an adverb or an adjective. However, even if we fix it to "Their performance is neck and neck", it still doesn't sit well with me. This expression comes from horse racing, and describes the position of horses relative to each other, not their performance.

So you can say:
They're neck and neck.

If you want a verb in there, it has to be some kind of a verb that describes motion. E.g. "They're running neck and neck". "Performing neck and neck" would sound strange because "perform" doesn't indicate movement, so it doesn't fit well with something that describes position.

Get rid of the “a”(and “quiet”). It can just be “Their performance is neck and neck”.