High-Q resonances excited by bound states in a continuum (BIC) in metamaterial have sparked increasing research interests for its prospective applications in sensing, lasing, and nonlinear optical devices. This paper introduces a quasi-BIC supported flexible terahertz (THz) metamaterial sensor for curvature measurement. The symmetry-protected BICs based on metallic rectangle structures on a thin flexible polyimide substrate are designed. After introducing defect perturbations along the x and y directions into the metallic rectangle, two observable quasi-BICs would be excited by x and y-polarized THz waves, respectively. Moreover, an experimental demonstration of the curvature sensing performance is conducted by monitoring spectral evolution in response to the bending BIC metallic metamaterial. The measured results indicate that the resonant transmission amplitude is sensitive to variations in the applied curvatures. The flexible, easy manufacturing, and highly-sensitive quasi-BIC micro structured metamaterial sensor paves a new way toward developing sensitive and wearable BICs-related optoelectronic devices.