Recent research has explored integrating lattice vector quantization (LVQ) into learned image compression models. Due to its more efficient Voronoi covering of vector space than scalar quantization (SQ), LVQ achieves better rate-distortion (R-D) performance than SQ, while still retaining the low complexity advantage of SQ. However, existing LVQ-based methods have two shortcomings: 1) lack of a multirate coding mode, hence incapable to operate at different rates; 2) the use of a fixed lattice basis, hence nonadaptive to changing source distributions. To overcome these shortcomings, we propose a novel adaptive LVQ method, which is the first among LVQ-based methods to achieve both rate and domain adaptations. By scaling the lattice basis vector, our method can adjust the density of lattice points to achieve various bit rate targets, achieving superior R-D performance to current SQ-based variable rate models. Additionally, by using a learned invertible linear transformation between two different input domains, we can reshape the predefined lattice cell to better represent the target domain, further improving the R-D performance. To our knowledge, this paper represents the first attempt to propose a unified solution for rate adaptation and domain adaptation through quantizer design.