Is it bad UX to have one background for the entire website except for one specific page?
I mean, would it be strange and lead to questions for users who are uncomfortable with this, or is it not a problem?
Consistency is always an important part of usability but that's not to say that some things can't change. It really depends on why you want the page to look different.
For example, it is common for a website to use colour coding to identify different areas of a website, where things like text, headings and links all change colour. That helps users to identify their current location.
If you look at a large and diverse website like the BBC, it uses different coloured backgrounds (and even different layouts) for its News, Sport, Weather and Children's sections. This is perfectly reasonable since each of those sections has its own content, tasks and audience.
Some pages of a site may also use different backgrounds to identify modal states like checkouts and registration pages.
First you should think about the purpose for a background change instead of starting by looking the reasons for not doing so. Does it add any value to the user? Is that distinction really needed?
If you want to change just one page you should have a good reason to do it, and you should take into account things like:
Absolutly isn't a problem! I undestand that the consistency is important in the design but with the the introduction of iOS 7 and 8 we have the possibility to introduce many background(including beuatiful gradients). If you have a webpage that need flow and user friendly you must have a very nice convination with awesome backgrounds and the user feels fresh and awesome for their eyes, and you can add some animations with the different backgrounds that you introduced.