We can certainly argue that Athenian culture in the golden age was a product of the democracy that Athens enjoyed. We can see this in a number of ways.
First, we can see this in Greek dramas. These dramas (like those of Aeschylus or Sophocles) were funded by the government. The reason for this is that they were supposed to help educate the people. This was important because Athens was a democracy that relied on an educated populace.
Second, we can look at history. Thucydides is seen as the first true historian because he wrote about the human causes of war and politics instead of ascribing them to divine causes. This is an attitude in keeping with a democracy in which the common people are a driving force. If the people are part of the government, it is important for them to know what sorts of things cause wars and other such events so they can better understand what to do.
Finally, we can look at art. Greek art of the time was focused on the human being as an object of beauty. This is an attitude that can be fostered by democracy. Since people are important to a democracy, it makes sense that democratic art would focus on people.
In these ways, we can say that much of Athenian golden age culture was a product of its democratic system.