In an expression that states, 'we will treat your work with the strictest confidence' does this have the meaning of 'with the utmost confidentiality'? Thanks.

Most likely yes, they're saying the same thing. However, because confidence does have multiple meanings what is important to look at is the context in which the statement is made.

"We will treat your work with the strictest discretion."

I believe that is what the first statement is trying to say. Using strictest with the other definitions of confidence doesn't make a whole lot of sense.

Best way to know for sure is how it's being used with the rest of the conversation or material. Context is vital to understanding the blurred lines that are all around us in the English language.