Since the discovery of high-order harmonic generation (HHG) in solids(1-3) , much effort has been devoted to understand its generation mechanism and both inter- and intraband transitions are known to be essential(1-10) . However, intraband transitions are affected by the electronic structure of a solid, and how they contribute to nonlinear carrier generation and HHG remains an open question. Here we use mid-infrared laser pulses to study HHG in CdSe and CdS quantum dots, where quantum confinement can be used to control the intraband transitions. We find that both HHG intensity per excited volume and generated carrier density increase when the average quantum dot size is increased from about 2 to 3 nm. We show that the reduction in sub-bandgap energy in larger quantum dots enhances intraband transitions, and this-in turn-increases the rate of photocarrier injection by coupling with interband transitions, resulting in enhanced HHG.