You can always clean the cartridge on the tank water to be disposed after a water change. But this depend on how good the cartridge components still are. Also, most of these cartridges include charcol which some people feel uneasy about leaving it on the filter for too long (there seems to be a debate on if it is harmless or not to keep charcol for long periods).
Anyhow, when replacing the cartridge just get rid of the old one, and before installing the new one rinse it with tank water (or treated water). The only bad thing with replacing is that you are loosing benefitial bacteria, that's why with bigger HOBs it is recommended to change 1 at the time. On smaller ones that only have 1 cartridge there is no option at some point you will need to replace it.
I was just looking at those aqueon filter cuz i am not familiar wiht them and they seem to have a media insert in the actual cartridge. So I guess you want to change just that internal part (or maybe customize it by using other media if you like... )
I guess a few members here have aqueons so they will be in a better position to give you options based on experience, my comment above is more about HOB filters in general