According to at least some example embodiments of the inventive concepts, an apparatus and method for encoding videos, a method for outputting a video quality index, and a video encoder are provided.The apparatus for encoding video includes a quantizer configured to, based on a size of an initial quantization parameter of input data, generate output data by quantizing the input data to increase an objective evaluation value of encoded data generated from the output data, or generate the output data by quantizing the input data to increase a subjective evaluation value of the encoded data.