Yes, if the veneer is unvarnished and untreated, you can stain it with a paint for wood. You’ll need to sand down the surface of the wood first, getting it smooth and rid of dust and wood flakes; once it’s sanded down, wipe down the surface with a very slightly dampened cloth to pick up remaining specks before applying the stain. Varnished veneers can be stained, as well, but will require a little more work in removing the treatment when it comes to sanding down – you may not be able to completely remove the coloring through sanding, but if you’re planning on staining over the veneer with a new, darker color altogether, then this shouldn’t be an issue, as the new treatment will cover and hide the old.