There are a couple of issues here.
Grounding the generator frame can be done via the Equipment Grounding Conductor
in the cable you connect it with (to the building electrode). Certainly driving
an additional rod when the ground thaws (~$10 plus about $5 worth of hardware)
is a safer approach but it will still have to be bonded back to the house via
the EGC.
The other issue is grounding the neutral. If your transfer equipment switches
the neutral you will need to ground the neutral in the generator.
In most cases the transfer switch only transfers the hot legs so you leave the
neutral isolated in the generator (as most portable generators are shipped).
This grounds the neutral via your main bonding jumper in the service panel.