A degree is definitely not a measure of intelligence. It is more a measure of ability. You have the ability of taking classes, learning the material desired, and passing exams. Intelligence is very complex, which is why true intelligence tests have multiple parts, there is no one thing that measures intelligence accurately.
That being said, I am sure there is a baseline level of intelligence one needs to be able to complete a degree program, however completing a degree doesn't mean someone is now more intelligent than someone without a degree.
What advantages a college degree does give is the idea that someone has studied that topic extensively for a few years, as well as studying various other fields. People with degrees in business for example, have studied various parts of business while also probably taking some English, math, science, and history courses. When they go to get a job, it is expected that this degree gives them the knowledge necessary to be better in the workforce than someone without the degree with the same amount of experience.
Now, there are always exceptions to this. Experience is typically held higher than education is. So someone with no degree and years of experience is typically held higher than someone with a degree and no experience.
To get back to the original question though, no, having a degree does not necessarily make you more intelligent. It is a mark of ability rather than intelligence, which is something that way more complex and inherent.
Now, when you delve into most graduate study is when you take an average student and make them turn a field inside out to explore it. This can lead to a higher intelligence within that field. This is why those with Ph.D.'s are viewed as experts in their field. I am working on a Ph.D. in Psychology, when I am finished I will be psychologist. Does that mean I am more intelligent than someone with no degree? Not at all. In fact, I'm positive there are people way more intelligent than me. However, it means that I have reached the level of study within psychology to be considered a professional expert in that field. I am in no way more intelligent than even someone who has just a BS in Psychology, though I would have a higher understanding of the field. This goes back to how truly complex intelligence is.