Is it possible to store flyback voltage in capacitors?
Yes, it is possible to store that energy in a capacitor; after all, that's exactly what flyback-mode switching power converters do! So your idea isn't silly at that level.

However, using that energy to "boost" the motor drive is a little misguided, because the amount of energy you get from the flyback effect (which is due to "leakage" inductance in the motor) is orders of magnitude smaller than the amount of energy required to accelerate the rotor to any signficant degree. In other words, the effort required to implement this would not be rewarded with any practical benefit.

You may be thinking about regenerative braking systems. But such systems are not storing "flyback" energy &mdash; they're actually using the motor as a generator and storing the energy created by physically slowing down the motor (and the attached machine or vehicle, etc.).