I have a set of direct current solar panels mounted on a metal structure (carport). All the panels and the structure are grounded to a ground rod driven near one of the support posts.

Can I use the same ground rod for the ground connection of an alternating current subpanel mounted on one of the support posts?

Yes, since if separate AC system is used nearby, and it has it's own ground rod, they must be connected either with a direct wire or through the common equipment ground system.

All ground rods need to be connected together.