In this paper, a new method for the construction of a quantum stabilizer code from circulant permutation matrices is discussed. First, we choose a finite-length vector randomly, and we can construct circulant permutation matrices from the vectors. Then, the parity-check matrix can be produce from the circulant permutation matrices. Hence, the generators of stabilizer code are determined according to the parity-check matrix and quantum stabilizer group are defined from the generators. From the stabilizer group, codewords of the proposed quantum codes can also be generated. Finally, a complete efficient encoding and decoding quantum circuit of [[7,1,3]] is proposed. [[7,1,3]] is stabilizer code that construction based on our method is an seven-qubit code that protects a one-qubit state with up to one error, which is very important for quantum information processing.