Can I use the grounding rods of a main panel as the grounding rods for my sub panel
It's not a separate building

Having a breezeway between the buildings means it is one building in the eyes of NEC.  You not only can use the same ground rods, you can simply use the ground wires you already ran.  

Further, you do not require a shutoff switch (typically implemented as a main breaker) in the garage's panel. Although if you're committed to it, it does no harm.