Photon emission with nonclassical photon statistics is discussed with a single InAlAs quantum dot. The deterministic single-photon generation in which the emitted photon wavelength matches well to the highly sensitive wavelength region of highly efficient, low-noise Si-single-photon detectors and also to an atmospheric transmission window is demonstrated. Competing transition processes between neutral and charged exciton species originating from an exclusive formation in the same single quantum dot are clarified. It was found that suppressing the charged exciton formation is possible by a quasi-resonant excitation for a deterministic monochromatic single-photon generation. Polarization-dependent photoluminescence spectroscopy clearly indicates the preservation of photon polarizations between photons emitted by biexciton/exciton recombinations. Furthermore, the deterministic polarization-correlated photon pair generation with biexciton-exciton cascaded transition occurring in a single quantum dot is directly confirmed by the polarized second-order photon correlation measurements. This indicates a longer polarization flip time than the exciton lifetime, which is an essential requirement for the deterministic Einstein-Podolsky-Rosen photon pair generation under the present biexciton-exciton cascaded transition scheme.