The cat above glows because it was exposed to a virus that carried a suitable gene:
US researcher Eric Poeschla has produced three glowing GM cats by using a virus to carry a gene, called green fluorescent protein (GFP), into the eggs from which the animals eventually grew. [...]
The GFP gene, which has its origins in jellyfish, expresses proteins that fluoresce when illuminated with certain frequencies of light. (citation)
Now that was about intentional human action, genetic modification, which is not what you asked about. But what I noticed in that article is that the modification to the egg is virus-borne. Ok, could that happen without human intervention?
This article from a veterinary site talks about diseases that can be transmitted to eggs (and the resulting young) in birds. Birds are not cats, but let's see what they have to say about vectors:
There are many infectious organisms that can be transferred from the hen to the egg that may cause the egg to die. In some cases, the infectious organism may infect the egg, yet the embryo may continue developing, and may even hatch, carrying the organism at hatch time. If an organism is passed from an infected hen directly into an egg, and then into the developing embryo, this is called vertical transmission.
We know that something similar happens in mammals; a mother who is infected, or drug-addicted, can transmit that condition to her fetus.
So, what we need is a way for a jellyfish-borne virus to affect a pregnant cat.
Fortunately, cats love fish, and fish are found on beaches, where jellyfish are also found. An infection through that vector to enough cats over time, and with the affected gene being transmitted to those cats' young and so on, could give you occasional glowing cats.