I suspect you're referring more to visual design, correct? If you mean design in the broader sense then comparing "Professional" design with "Non-Professional" is like asking if you can test the difference between a product made by someone who knows how to build a product versus someone who doesn't. Anyone building a product is, in fact, designing. Whether they are good at it is another thing entirely. Design is combination of "how things work" and "how things look" (and I'd even argue the latter is still a child of the former).

Visual design as decoration is probably less effective, and I think you'll get a lot of that with "Unprofessional" designers. Visual design as a way to amplify/enhance existing well-designed interactions is definitely effective (i.e. strong button affordance, good use of color to convey meaning, etc). This is the type of output you'll get with a "Professional" designer.

Philosophy aside, I think you'd probably do better with qualitative testing. Do people tend to complete tasks easier if the action being tested "looks" easier? Does the brand identity match the audience you're targeting? Are your design patterns consistent across the site? Etc, etc. It's certainly a bit more nebulous but I think with the right questions you could certainly use A/B testing to answer some of these things.