If you haven't already, inspect each of the fans and look for cables touching. Besides that, I would do as others suggested and download a monitoring software to pull fan speeds and system temps. It sounds like one of the fans may be damaged causing it to get unbalanced or something and make more noise. I would try to reproduce the issue and see if it only occurs if certain fans spin up - many gpus and cpu coolers will have a zero rpm mode which was be the reason it is intermittent. Run something like furmark to increase your gpu temps and cinebench r23 to increase your cpu temps. If you are comfortable taking apart your pc, you could also unplug the fans with the side panel open and see if the noise appears again, then narrow down the issue by slowly testing and reattaching fans. It could also be an issue with the power supply / power supply fan as its a prebuilt and psus are a place they will cheap out.
If it is one of the case fans, make sure it is not an issue with a certain header rather than the fan bugging out by trying the affected fan in a different header.