Is it possible to calculate a star's apparent magnitude in another photometric band knowing its visual magnitude? Or, in other words, is it possible to know a star's luminosity in a certain photometric band?
Proxima Centauri's Wikipedia page provides the apparent magntiude in each UBVRIJHK band (getting progressively brighter) as well as the bolometric luminosity and the luminosity in visual wavelengths.
But this amount of information seems limited to Proxima Centauri and a few select other stars.
Is there a way to calculate a star's apparent magnitude/luminosity in another photometric band knowing its visual apparent magnitude/luminosity? Other factors like distance, size, and mass are also known.
No there isn't if you know nothing else about the star. Knowing the distance doesn't help. Knowing the mass and size of the star (how would you know these without other sources of information?) allows you to estimate what sort of a star it is and therefore you have some idea of the spectrum of that kind of star. Given that as a template, then it is possible to estimate what colour the star is an hence how bright it would be in any other photometric band.
e.g. If you have the temperature then you should find a table of temperature vs whatever colour you are interested in - so B-V if you want the B-magnitude. You interpolate on the table, find the B-V and add that to your known V to get B.
This assumes there is no reddening/extinction involved. If there is then your estimate will be wrong (to small for bands bluer than V, to large for bands redder than V).
A suitable interpolation table for main sequence stars can be found at
https://www.pas.rochester.edu/~emamajek/EEM_dwarf_UBVIJHK_colors_Teff.txt