The only decrease in quality will be the likely performance difference between the Full Frame and APS-C sensor. In general FF lenses used on APS-C cameras perform slightly better than when used on FF cameras because the edges of the light circle cast by the lens, where most of the weaknesses of a lens are most obvious, are cropped off by the smaller sensor. Using a FF lens on a crop body in no way degrades the quality of the lens.
When viewed at the same display size, photos taken with a smaller sensor are magnified by a larger factor than photos taken with a larger sensor. Any imperfections in the lens' performance will also be magnified, but that is just as true if you use a lens designed only for APS-C cameras as it is if you use a lens designed for FF cameras.
And taking into account the dismally poor optical performance of the EF 75-300mm f/4-5.6, the EF 24-105mm f/4 L IS will be a considerable upgrade for use at 70mm and longer than what you are using now. Because the hood of the 24-105 is designed to allow a 24mm Field of View (FoV) with a FF sensor, you may need to shade the lens from strong off axis light sources just out of the frame when using it at longer focal lengths to prevent flare. Of course this is true of any zoom lens when used at the longer focal lengths in its range.
The EF 24-105mm f/4 is a wonderful lens if you need the full range of its available focal lengths. And it is built like a tank. Mine has taken years of abuse and still performs as well as it ever has. But if you only plan to use it at 70mm and up, you might consider the EF 70-200mm f/4 L or EF 70-200mm f/4 L IS instead. You will get better optical performance in the common focal length ranges between the two lenses and you will get a lens that goes all the way out to 200mm. The non-IS version is about $250 less than the 24-105, the IS version is about $250 more than the 24-105.
The "megapixel reduction" to which you refer in your question is applicable if you shoot with a FF body and then crop the image to give the same FoV as you would have obtained by using an APS-C body. It has absolutely nothing to do with using an EF lens on an APS-C camera.