I think the answer is "No", but it depends on where you draw the line at "trivial" and "programming language." I think ANSI SQL isn't Turing complete. But I don't consider that a programming language. I think all modern general purpose programming languages are Turing complete.

That is, programs from general purpose programming languages can be re-written in any other Turing complete language. Not to be confused with the Zed Shaw "joke" that a programming language is not Turing complete if its interpreter does not run source code written in another Turing complete language, completely unmodified.