Honestly I experience the same thing very often. Filters are just really delicate sometimes and even if you are careful when removing media it can stir things up and when the filter is turned back on it spits out some solids. I find this to be most prevalent when dealing with power filters that have the mechanical filter pad behind the biological sponge, like Whisper filters. When removing the mechanical pad, despite how delicate you do it, it can still disturb the bio sponge a little. And also filters that contain bio media that you can put in mesh bags like, AquaClear, I have noticed that when removing it for filter cleaning, I usually soak it in a small bucket with tank water. Even as I gently take it out and put it in I notice particulates floating around. So when it comes time to put it back in, the same thing happens when I fill the filter back up, and I can see debris floating around.
Periodically, usually I try to do it 2 times a month, the filter should be completely removed from the tank for a more thorough cleaning of the inside, tubes, and impeller. This requires you to take out all media, which I would suggest soaking your biological media. On other occasions when the mechanical pad needs rinsing, you don't have to remove the whole filter if you don't want to. I suggestion is to make sure you rinse the mechanical pad in tank water to ensure the bacteria in there don't get removed/die.
A little trick I do, (well not so much a trick...) I have a few different fish nets, and what I do is that prior to turning power filters back on, I place the net where the outtake is so the waterfall falls completely in the net, and that actually catches most of the solid debris. I don't necessarily do this all the time, because the stuff that comes out isn't necessarily bad and will eventually get sucked up by the filter. But I understand that it can be a nuisance when just after you cleaned your tank and filters, you turn the filters back on, and there's stuff floating around in the tank.