Three supergiant flares from soft γ-ray repeaters are observed, with typical released energy of ∼1044–47 erg. A conventional model (i.e. the magnetar model) for such events is catastrophic magnetism-powered instability through a magnetohydrodynamic process, in which a significant part of the short–hard γ-ray bursts could also be the result of magnetars. Based on various observational features (e.g. precession, glitches, thermal photon emission) and the underlying theory of strong interaction (quantum chromodynamics), it cannot yet be ruled out that pulsar-like stars might be actually solid quark stars. Strain energy develops during the life of a solid star, and starquakes could occur when stellar stresses reach a critical value, with a huge amount of energy released. An alternative model for supergiant flares of soft γ-ray repeaters is presented, in which the energy release during a starquake of a solid quark star is calculated. Numerical results for spherically asymmetric solid stars show that the gravitational energy released during a giant quake could be as high as 1048 erg if the tangential pressure is slightly higher than the radial one. Difficulties in magnetar models may be overcome if anomalous X-ray pulsars/soft γ-ray repeaters are accreting solid quark stars with mass ∼1–2 M⊙.