change background color on just ONE page

JanWillem_Aikens​ 

You can simply create a shaded rectangle set to be behind text. This will be anchored to text though, so if the text moves to a different page, the background will as well.
Size the rectangle to fill the entire page.

This does not need to be in the header, but is subject to being dislocated by moving text.

Word Doesn't Know What a Page Is by Daiya Mitchell, MVP( https://wordmvp.com/Mac/PagesInWord.html)
That was written some time ago, and is still true.

As far as I can tell, the page color setting under the Design tab is for the entire document.

Logged in just to say this doesn't only seem complicated, but unbelievable this isn't just possible out of the box.

JensP220 You will need to insert a Next Page Section Break before and after that page and in the Header for that Section, uncheck the Same as Previous option and Insert a Rectangular Shape that fills the page and is of the colour that you want.  It should be set so that it is "Behind the text" and you will need to unlink the Header in the following Section so that the shape can be removed from that Section.