Here is your data:
wavelengths =
Import["http://web.lemoyne.edu/giunta/balmer.html", "Data"][[1,2 ;; 5]][[All, 2]]
giving
{6562.1, 4860.74, 4340.1, 4101.2}
You say this corresponds to m from 3 to 6.
So let's fit it to your expression with h as the unknown:
FindFit[Transpose[{Range[3, 6], wavelengths}], h m^2/(m^2 - 4), h, m]
Giving:
{h -> 3645.6}
