My cousin has already used acrylic paint on wood to produce paintings such as these:
But his paintings are not "glowing", i.e. they don't have bright colors. He asked me how to make them bright after the painting is done.
So, my cousin is looking for solutions such as these:
A bright "glowing" sample is provided by my cousin below:
I think it's hard to judge the real difference between the paintings your nephew made and the example of the goal he has in mind: the first are taken in what seems a yellowish light, while the second is colour corrected (at least, it seems to have a proper white balance set) and taken in bright light.
Assuming both the eyes of the giraffe and the fur on the hat of the bird are white in reality, this reproduction comes closer to what the artworks look like:
The visual change this inflicts notwithstanding, there are a few remedies to make the colours in the paintings brighter (assuming no coat has been applied afterwards, since that potentially influences the solutions and makes them dependent on the nature of the coating; if this is the case, let me know, and I can update my answer):
Other answers point out the role the support plays in the saturation of colours. Wood, especially dried, porous, and/or unprepared, may indeed absorb part of the paint or water in acrylics. Apart from their colours becoming brighter, this is another good reason to use undiluted acrylics, as they will effectively form a plastic layer on top of the support. It then functions similar to acrylic gesso, a property on which some of the solutions offered here are based.
Technically this is not an answer, because I don't know how to make colors "glow" after the painting is finished. My best bet is to try a glossy varnish.
The main reason why the paintings don't glow is that they don't reflect enough light. Two main causes for that come to my mind:
The wood beneath the paints absorbs light. He should prime the wood with white before painting the pictures on it. That way the light that travels through the layer of colored paint is reflected by the white layer instead of partially absorbed by the wood.
He uses the wrong paints. The physical an chemical composition of paint decides how it looks. Glossy paint glows more than matte paint. Pastel colors tend to seem shinier than dark colors. Earthy tones don't ever seem to glow at all. Cheap paints tend to be more muted or chalky and some paints in really powerful shining colors are very expensive.
The only way to make his paintings "glow" is to paint them with paints that "glow". Which brings me to my final question:
Does he compare apples with oranges?
The example picture seems to be a porcelain plate, which already is white and smooth and very reflective. I assume the image was painted with porcelain paints or pens, which tend to be semi-transparent. Those colors "glow" because the incoming light can pass through the color layer, is reflected off the white porcelain surface, passes through the color again and then reaches your eye. That means that a relatively small percentage of light is absorbed by the object and a high percentage of the light gets reflected.
Your cousin's pictures are painted with matte, opaque colors on a rough, natural surface. The light cannot pass well through the color, but is reflected by the upper surface of the paint. Under a microscope you could see that the paint itself has a rough surface. Due to this roughness some light gets absorbed by the paint and some is scattered away from your eyes. In the end, less light reaches your eye and the painting doesn't "glow".
The physical properties of these two types of paint are so extremely different that you cannot realistically compare them. If he wants "glowy" colors like in the example, he would have to prime the wood with a glossy white and then paint the images using transparent or semi-transperant paints like ceramic paints...
You used the term "glow", but as others have pointed out, what you're referring to is making the colors pop; making them more vivid and with more depth ("glowing" requires that they emit their own light, which can be done but that's not what you're looking for here). Even if your cousin did make the colors literally glow in the second example, we wouldn't be able to see that effect in a posted image.
The answers by Joachim and Elmy do a good job of explaining how to make the colors pop when starting from scratch, and why the "good" example in the second picture was able to accomplish it while the items in the first picture didn't.
Your question focuses on what can be done after the fact to make items like the ones in the first picture pop. The vividness and visual depth of the color is affected by a number of things.
The substrate. As Elmy describes, a big factor is the base on which the painting is created. Ideally, that should be smooth and bright white. Photographers sometimes make prints pop even more by printing on "metallic" paper. At this point, it's too late to do anything about the substrate.
The paint. It's too late to start with paint that's better for this purpose. The existing designs could be touched up or repainted with better paint or tinted glaze, but that sort of defeats the premise of the question -- starting at the point that the existing paintings are done. Trying to touch up or repaint risks messing up what's there; it would make more sense to just start over and use the right materials to accomplish the objective.
Coating or finish. The question asks about a spray or solution that can make the existing colors pop more. Elmy's answer suggests applying a glossy varnish. That will improve it, and there are many clear finishes you could use. A popular one is Crystal Clear Acrylic spray finish by Krylon, which doesn't yellow. But you can do better than the improvement you would get with a thin, glossy finish.
This starting point actually isn't too different from a common crafting task of making jewelry pendants, coasters, and similar items from small paintings or pictures clipped from printed material. The artwork often suffers from not having vivid colors that pop in the original, but the finished product is greatly improved in that respect. The key is embedding the art in a clear resin that is well bonded to the art's surface. The resin "cover" fills any irregularities on the artwork surface, and is thick enough so that the top surface is smooth. It acts somewhat like a lens, improving contrast and clarity. There are a number of ways this can be done:
Using a pre-formed glass or plastic dome. This works best with something that is perfectly flat, like something printed on paper. The paper is saturated with a clear adhesive and bonded to the flat backside of the dome so that the dome makes good optical contact with the surface. This approach wouldn't be suitable for artwork on wood with an uneven surface.
Embedding in clear resin. Two types of resin are often used for this approach. One is two-part resin designed to be poured over a surface as a clear top coat. It flows over the surface to form a thick, smooth coating that hardens into a clear plastic layer. Another is clear embossing powder, which is melted in a temperature-controlled pan and then poured over the artwork. Controlling the temperature is important because it can discolor a little if overheated (an effect some people like because it provides an "antique" appearance).
Covering with a "dimensional" finish. There are a number of typically water-based products designed to be poured over the surface in a fairly thick layer, or series of layers, that dry to form a relatively thick, clear coating. They are viscous enough, and have enough surface tension, to generally stay where you put it, even with a thick puddle, but it self-levels to a smooth, uniform surface. The result is similar to resin but without the mess, fumes, waste, or need for special equipment; you just dispense what you need from a bottle and let it dry.
I'm not sure how impervious to water these products are when dry. Popular products include JudiKins Diamond Glaze, Mod Podge Dimensional Magic, and Ranger Glossy Accents. If you do a search on "dimensional medium", you'll find lots of suitable products, and YouTube videos comparing them. Some of the alternatives may yellow a little over time.
