Yes, one could manufacture a reason why this would work in a universe similar to ours (in non-laboratory settings). But, no, for the reasons that others have pointed out, it’s not possible in real life.
One thing others have overlooked is that moonlight is not merely dim sunlight; the sunlight has interacted with the moon rocks, and nonlinear or filtering processes could have taken place. So...
The mechanism is a mineral which occurs naturally on the moon’s surface, but on Earth it is confined to the core and inaccessible in any meaningful quantity. This mineral has a unique spectral emission line in the ultraviolet, labeled $\Xi$, which is remarkably narrow for a solid-state material. This material absorbs bright visible sunlight via multi-photon absorption and subsequently emits this narrow line of UV light with a purity and brightness unparalleled on Earth except in a synchrotron or a specialized laser lab.
Enterprising chemists have discovered/invented a molecule, $\Gamma$, which is unique in its complementary absorption line, $\Xi^*$, leading to broadband visible photoluminescence upon excitation by moonlight. But most remarkably, $\Gamma$ has a long-lived alternate electronic configuration, and the absorption of light with energy even a fraction above $\Xi$ is sufficient to bump $\Gamma$ into this quasi-stable state, which is dark (i.e. no photoluminescence). Over the course of a few microseconds, $\Gamma$ relaxes to its ground state by non-radiative means, ready for excitation once more. Thus, broadband excitation leads to orders of magnitude dimmer visible luminescence compared to excitation directly at $\Xi$. You could probably image a message written in $\Gamma$ in your lab using a bright UV lamp, some specialized filters, optics, and a fancy electron-multiplying CCD camera (but it won’t fit in your pocket!).
Then, as a practical matter, there are simply no easily accessible sources of light which have a narrow enough emission at the ideal wavelength to excite photoluminescence in $\Gamma$ without quenching it with higher-energy light. None, that is, except the moon.