I don't believe you need additional filtration if your set up is proportioned correctly. However I am a newbie as well and would suggest you consider some additional responses to be sure.
I didn't think I needed media filters either but my fish tank got dirtier and dirtier from "something" from the grow beds and, eventually, a bunch of my fish died. Horrible. I have added some filters now in the form of nylon stockings at the bottom of the siphon pipes and that has helped considerably. I also put a filter in the line from the sump tank to the fish tank. That has been a very mixed experience.The first filter, a carbon one, clogged in about an hour and I nearly pumped all the water out of the fish tank. Then I put in a paper filter and that needs to be cleaned daily. So I'm working towards a better solution. (and yes I'm new at this also -- only 3 months in). The bottom line is to watch the water in your fish tank carefully. Conventional wisdom says no filter is needed since the media are a biofilter but my experience wasn't like that.
It will all depend on the type of grow bed you are using. See if you have gravel beds and you have enough gravel beds for your fish load and you are circulating enough of your water through those gravel beds each hour, then NO you would not need more filtration.
However, if you have a raft system and your grow bed is just a bed of water with a foam raft over it. Then Yes, you might need some media filtration. There have been some experiments with very low stocking and raft beds with no additional filtration but the water quality and condition of plant roots needs to be watched closely to ensure you are not building up gunk to the point of a problem.