Does a UPS charge it’s battery even if the computer is turned on

Does your UPS also properly protect against surges as well as power drops? If not it could be that the power outage is preceded by a brief surge which rather then smoothing the UPS reacts badly too (or allows through and the PC's PSU reacts badly to it). I would expect that must UPS devices are also designed to smooth out surges as well as protecting against outages, but it might be the case that either not all do, or that the power problem your incoming supply presents is outside the range of what your UPS is intended to deal with. It could also indicate that your USP is faulty and not correctly dealing with circumstances that it should be able to brush off.

It would be a good idea to add the make+model of UPS to your question. Also, the country+area you are in may make a difference to how likely certain electricity supply anomalies are, so you might want to specify that information too.

The only sensible answer is: It depends.

All UPSes I've seen so far don't produce any noise when they aren't supplying power to a computer, but then again none of them have fans. Also a thing to consider is that internal electronics (not fans) may produce noise while UPS is powering the computer and that some UPSes have buzzers which can get quite annoying, especially if there's no way to turn them off. Also you may hear noise from relays when UPS switches between its internal power source and external power source. In some cases that can get quite annoying, especially if it so happens that your mains voltage is near UPS's tripping point. This can also lead to reduced relay lifetime.

UPSes usually use lead-acid batteries that are often sealed and they don't produce much heat when being charged and the charging circuitry itself, if it's slow enough, will not produce much heat.

So the bottom line is: I wouldn't expect a UPS to require fans when charging batteries or providing power to the computer from an external power source, but if you want to be safe, ask the manufacturer.