Actually, you CAN use factor analysis to answer your question, but it's not quite so straightforward. Basically, there are two kinds of factor analysis: exploratory factor analysis (EFA) or confirmatory factory analysis (CFA). Exploratory factor analysis is used as you might expect - to identify which questions might be redundant and can be removed, and how the questions group together into an underlying structure. Confirmatory factor analysis is used to test whether the data gathered fits to an existing model.
When you say that you "think" the questions group together, it looks like you mean that they have some theoretical connection to each other. Remember, though, that factor analysis will simply look at how to explain variance in the responses, and so your theoretical factors might not line up at all with the actual factors in your study.
Generally, people field a larger survey of questions developed through a robust process, and use exploratory factor analysis to examine the latent structure of the instrument. Confirmatory factor analysis is more useful when prior EFAs have been done on an instrument - for instance, if we are checking how an instrument fares in a different population.
So, I would recommend doing some EFA work on your responses to see how things group. It might be that your four sections are about different things, but that the underlying factor structure cuts across your explicit domains. But since you don't really have an existing model (beyond "these questions go together"), CFA isn't really a good fit for your project.