How to find support beam on the second floor

Are you thinking you need it over a beam/support wall because of the weight?
Or are you asking how to find a floor joist?

Also make sure of it's measurements and that it will fit inside the closet.

I have worked on many homes where the spacing/location of the joists from 1st floor to second floor was different.
The idea of finding a floor register, they 'usually' mount to a joist side, so you find the joist, then calculate from there.
'usually' 16" but some are 19.2" and some are 24", and they can be altered for things like bathroom plumbing, chimneys and such.
Start with the 16" calculate over and find where you hope it to be.
Pull of baseboard, drill small hole near wall, which baseboard will cover. If you don't hit a joist no big deal. Try the other measurements. Or put a wire in the hole and feel over till you hit a joist.