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...