This is not a 'productive' grammar. There are certain cases (e.g. 近い・近くの、多い・多くの) where there are both noun and i-adjective forms, but you don't generally see "高くの". Where the noun form exists it will generally have a dictionary entry as well. And of course, there are only a few basic colours which even have the i-adjective form.
For the colours, I'm referencing from 色彩名詞と色彩形容詞の対立 (pdf) which is basically a corpus study of the usage of colour words.
Their hypothesis was that the noun forms are unnatural when used to describe "natural" (non-artificially coloured) things.
They took a large corpus (text from newspapers and literature), stripped out things like set phrases (赤の他人), names of places and titles of movies, and terms that were overrepresented in the data set (there was a newspaper column title, 青い山脈). They also removed compound words (青白い, 赤茶色の
), cases where the noun form was preceded with an adjective (深い青の), and cases like 赤や黄色のミニトマト and 赤青黄のおはじき referencing multiple colours.
Then they split the remaining cases by what was being described:
- Natural things (things that are not artificially coloured), including things like hair/eyes, sea/sky, flowers/birds
- Artificially coloured things (particularly clothing)
- 2D objects e.g. "line" or "letter" (文字)
- Unspecified (generic terms like もの and 部分)
- Colourants (ink, dye)
The colours they looked at were 白、黒、青、赤、黄色 and 茶色. In the texts they looked at, the adjective forms were much more common than the noun forms for the first four, 黄色い was more common than 黄色の but not by as much, and 茶色の was more common than 茶色い by more than 3:1.
When it came to the five types, the noun forms were primarily used to describe things in their "artificially" coloured set, not natural objects. When they split it by colour, though, 黄色の and 茶色の are used for natural things.
There's a lot more in here and on page 21 there's a sort of table summarising the differences - basically the adjective forms are natural, more vague, "analog" and the noun forms are artificial, more precise/scientific, "digital" (the analog/digital thing is their analogy).
I don't think this covers everything because they've specifically removed cases like "深い青の" and "青と赤の" where the noun versions might be more in use.
As mentioned in the comments, google counts are only estimates, and can be skewed by various other things (most of the first hits I get for 白の鳥 are about a book, 白の鳥と黒の鳥).
Just to add something to the pot, there's also sometimes specific additional noun forms, such as 青空.
For the 近くの vs. 近い example, there are differences in usage and in meaning (or range of meaning). For example, when talking about places, you would normally use 近くの when it appears at the start of a sentence.
And then there are specific cases where 近い has meanings that 近く doesn't, because 近く tends to be only used for physical distances. A specific example:
近くの親戚 (relatives who live close by)
近い親戚 (relatives who are close blood relations)