Variable-rate coding schemes that support a variety of different rates while maintaining the same fundamental encoder/decoder architectures are of great interest in practical communication systems. Similar error-correcting code schemes are also in demand in quantum settings. However, generally, it is difficult to introduce the variable-rate coding schemes into the quantum coding domain attributed to two challenges: Obtaining new quantum codes from old ones systematically and sharing encoder-decoder components among the newly obtained quantum codes. In this correspondence, a multiple-rate coding scheme is introduced into the quantum coding domain, we achieve this goal by providing a systematic method to construct new non-homogeneous quantum LDPC codes of CSS type from an old one (named as the mother code) whose classical parity-check matrix (PCM) is row-circulant. The basic idea is to split rows of the highest-rate stabilizer elaborately to produce the stabilizer for lower rates, which ensures that the generated code family owns the same code length. More importantly, thanks to the nested stabilizer structures possessed by the resulting quantum codes, our method also enables a scalable stabilizer circuit to be shared among them.