Perhaps you can clear up some confusion, because it appears that you are installing a 125amp subpanel yet only protecting it with a 60 amp breaker in the main?
Presuming this is a misprint and you are installing a 60 amp subpanel protected by a 60 amp breaker, individual #4 conductors should be run in raceway in all locations (even through studs) OR run as a cable assembly which can be run unprotected even when surface mounted.
You can run the cable through the 1 1/4″ holes but you will need protection plates if the edge of the hole is within 1 1/4″ of the edge of the studs.