Sounds like bleaching from something. You need to clean the area to get whatever it is that caused the bleaching out before you add anything or it may just bleach out too.

Once you get it really clean, then you can try to dye it. I'd want to try something that gives you more control than a spray dye (which, short of an airbrush, I've not heard of). What comes to my mind would be a brown Sharpie or a fabric pen -- maybe a brush on fabric dye. For the Sharpie, you might find an office supply or art supply store that will sell colors individually, but otherwise, they come in the multi-color packs -- like crayons. Look in a crafts store for fabric pens. You want dye, not a paint that will sit on top of the carpet. Try it on a scrap or rag of white cotton fabric to test the color first, and if you're happy with that, I'd start as deep into the nap -- as close to the base as possible and go slow and easy. If the color isn't working, you can stop before you get to the surface.

The pros probably have some other tools in their arsenal, but that's what I'd try first, especially if it is or will be covered by furniture. Good luck.