Brain-inspired hyperdimensional (HD) computing models cognition by exploiting properties of high dimensional statistics-high-dimensional vectors, instead of working with numeric values used in contemporary processors. A fundamental weakness of existing HD computing algorithms is that they require to use floating point models in order to provide acceptable accuracy on realistic classification problems. However, working with floating point values significantly increases the HD computation cost. To address this issue, we proposed QuantHD, a novel framework for quantization of HD computing model during training. QuantHD enables HD computing to work with a low-cost quantized model (binary or ternary model) while providing a similar accuracy as the floating point model. We accordingly propose an FPGA implementation which accelerates HD computing in both training and inference phases. We evaluate QuantHD accuracy and efficiency on various real-world applications, and observe that QuantHD can achieve on average 17.2% accuracy improvement as compared to the existing binarized HD computing algorithms which provide a similar computation cost. In terms of efficiency, QuantHD FPGA implementation can achieve on average 42.3× and 4.7× (34.1× and 4.1×) energy efficiency improvement and speedup during inference (training) as compared to the state-of-the-art HD computing algorithms.