The quasi frame is more efficient than the Frenet frame in investigating surfaces, and it is regarded a generalization frame of both the Frenet and Bishop frames. The geometry of quasi-Hasimoto surfaces in Minkowski 3-space ${\mathbb{E}}_1^3$ is investigated in this paper. For the three situations of non-lightlike curves, the geometric features of the quasi-Hasimoto surfaces in ${\mathbb{E}}_1^3$ are examined and the Gaussian and mean curvatures for each case are determined. The quasi-Hasimoto surfaces in ${\mathbb{E}}_1^3$ must satisfy a necessary and sufficient condition to be developable surfaces. As a result, the parameter curves of quasi-Hasimoto surfaces in ${\mathbb{E}}_1^3$ is described. Thus, the $s$-parameter and $t$-parameter curves of quasi-Hasimoto surfaces in ${\mathbb{E}}_1^3$ are said to be geodesics, asymptotic, and curvature lines under necessary and sufficient circumstances are proved. Finally, quasi curves and associated quasi-Hasimoto surface correspondences are discussed.