As a new color image representation tool, quaternion has achieved excellent results in the color image processing, because it treats the color image as a whole rather than as a separate color space component, thus it can make full use of the high correlation among RGB channels. Recently, low-rank quaternion matrix completion (LRQMC) methods have proven very useful for color image inpainting. In this article, we propose three novel LRQMC methods based on three quaternion-based bilinear factor (QBF) matrix norm minimization models. Specifically, we define quaternion double Frobenius norm (Q-DFN), quaternion double nuclear norm (Q-DNN), and quaternion Frobenius/nuclear norm (Q-FNN), and then show their relationship with quaternion-based matrix Schatten-p (Q-Schatten-p) norm for certain p values. The proposed methods can avoid computing quaternion singular value decompositions (QSVD) for large quaternion matrices, and thus can effectively reduce the calculation time compared with existing (LRQMC) methods. Furthermore, we also analyze the convergence of the algorithms, and introduce a rank estimation method for the quaternion matrix. The experimental results demonstrate the superior performance of the proposed methods over some state-of-the-art low-rank (quaternion) matrix completion methods.