I'm afraid your code might have come out a bit garbled.
It looks as if you are using XPath within some programming language where you are writing the XPath expression as a string literal in double quotes, and escaping the double quotes using the conventions of that host language. I assume also that you are using XPath 1.0.
Firstly, let's see what the XPath expression should look like:
Value[@Name=concat("Test ' both ",'" quotes"::LGC')]
Then, to put that in a string literal in your host language (if you haven't got some mechanism like C#'s verbatim string literals), you put a backslash before each quotation mark, and put the whole thing in quotes:
"Value[@Name=concat(\"Test ' both \",'\" quotes\"::LGC')]"