Will it show as blue color under dark UV light (say 390 nm to 400 nm range) ? Does it degrade over time under long exposure of UV light (i.e. losing its fluorescein characters) ? What specifications I need for my UV to fluoresce MB ?
According to this article, methylene blue is fluorescent:
Article Bile salt induced solubilization of methylene blue: study on...
Absorption and emission spectra are shown in that paper.
One of its excitation peak wavelengths is in the UV, at wavelengths shorter than 350 nm. It's emission peak is in the red region of the spectrum. Its fluorescence will probably be quite weak if you use 390-400 nm excitation, because the compound has very weak absorption in that range.
If you want to excite MB with UV light, you should use short-wave UV because the excitation peak is at around 290 nm.
The fluorescence probably will decay over time upon exposure to short-wave UV light due to photochemical reactions.
In addition to the paper shared by Prof. Shapiro, you could also go through the following to understand the mechanism of discoloration of Methylene Blue. The paper aims at explaining the effect of UV and visible irradiation on the pi-pi* and n-pi* transitions corresponding to the absorption of 293nm and 664nm by MB.