Material that absorbs only in NIR and emits in visible
Such materials not only exist but are common

Most people will have seen an actual demonstration of materials that exploit the phenomenon of a substance that absorbs NIR light and emits visible light. Most Green laser-pointers are based on their use. 

These usually rely on a semiconductor laser outputting near-infra-red (NIR) light at 1063nm wavelength which then passes through a crystal of KDP (mono potassium phosphate) or KTP (potassium titanyl phosphate) which emits green light at 532nm (which is the second harmonic of the original laser frequency). The mechanism involves non-linear optical processes (which usually require a fairly high intensity of light so won't happen as easily as phosphorescence, for example).