You're going to want to override vim's normal syntax highlighting for sqlString
.
I was able to get the behavior I think you're looking for with just this:
:syntax region m4String start=/`/ end=/'/
To achieve nesting, we have to tell vim that a m4 string can contain more of the same:
:syntax region m4String start=/`/ end=/'/ contains=m4String extend
If you'd like to color those strings separately instead of just disrupting the normal highlighting you can separately link to the String highlight group.
:hi link m4String String
I wasn't entirely sure what you were looking for though; if the backtick-apostrophe delimited strings are inside of the normal SQL strings this won't work.