Some filters have carbon cartridges that need to be replaced, I honestly don't like carbon for anything other than removing old medications, so I see carbon cartridges as useless for a betta's permanent home.
As jstn correctly said, the media in your filter should never be replaced, that is where all the beneficial bacteria established during your aquarium's cycle lives. You should also never rinse the filter media in tap water, but instead if you so wish you can take it out, give it a swish around in old aquarium water once every few months and put it back. This'll get rid of the larger debris whilst keeping your cycle in tact.
"There will come a time when three words uttered with charity and meekness shall receive a far more blessed reward than three thousand volumes written with disdainful sharpness of wit."