Firefox, Safari, and Chrome (as of late 2013) all do proper colour management - but Safari and Chrome colour manage only if the image contains an embedded profile. Otherwise they don't do any colour management.
Firefox is the same by default, but you can change that to colour-manage even images with no embedded profile (it assumes that images with no embedded profile are sRGB, a fair guess). It does this if setting "gfx.color_management.mode" is 1 (default is 2). You get at this setting (in Firefox) by entering the URL "about:config" ( i.e. enter about:config in the URL bar instead of www....)
It sounds as if either the images you are looking at don't have embedded profiles, or the new web site you are using strips out embedded profiles (perversely, some web sites do that - I assume it's because the web site managers are colour-blind). The NEC PA241W is wide gamut, so without colour management sRGB colours are too vivid.
(IE - even IE10 - screws up colour management, which won't work - can't be made to work - on a wide gamut monitor such as the PA241W.)