LSQ14bdq and SN 2006oz are super-luminous, hydrogen-poor, SNe with double-humped light curves. We show that a Quark-Nova (QN; explosive transition of the neutron star to a quark star) occurring in a massive binary, experiencing two Common Envelope (C
E) phases, can quantitatively explain the light curves of LSQ14bdq and SN 2006oz. The more massive component (A) explodes first as a normal SN, yielding a Neutron Star which ejects the hydrogen envelope of the companion when the system enters its first CE phase. During the second CE phase, the NS spirals into and inflates the second He-rich CE. In the process it gains mass and triggers a Quark-Nova, outside of the CO core, leaving behind a Quark Star. The first hump in our model is the QN shock re-energizing the expanded He-rich CE. The QN occurs when the He-rich envelope is near maximum size (~ 1000R_sun) and imparts enough energy to unbind and eject the envelope. Subsequent merging of the Quark Star with the CO core of component B, driven by gravitational radiation, turns the Quark star to a Black Hole. The ensuing Black Hole accretion provides sufficient power for the second brighter and long lasting hump. Our model suggests a possible connection between SLSNe-I and type Ic-BL SNe which occur when the Quark Nova is triggered inside the CO core. We estimate the rate of QNe in massive binaries during the second CE phase to be ~ 5x10^(-5) of that of core-collapse SNe.
