Brain-inspired Hyperdimensional (HD) computing is an emerging technique for low-power/energy designs in many machine learning tasks. Recent works further exploit the low-cost quantized (bipolarized or ternarized) HD model and report dramatic improvements in energy efficiency. However, the quantization loss of HD models leads to a severe drop in classification accuracy. This paper proposes a projection-based quantization framework for HD computing (PQ-HDC) to achieve a flexible and efficient trade-off between accuracy and efficiency. While previous works exploit thresholding-quantization schemes, the proposed PQ-HDC progressively reduces quantization loss using a linear combination of bipolarized HD models. Furthermore, PQ-HDC allows quantization with flexible bit-width while preserving the computational efficiency of the Hamming distance computation. Experimental results on the benchmark dataset demonstrate that PQ-HDC achieves a 2.82% improvement in accuracy over the state-of-the-art method.