How do I update GRUB in Arch Linux

The update-grub command is just a script which runs the grub-mkconfig tool to generate a grub.cfg file. See the Archlinux GRUB documentation. It refers to the following:

# grub-mkconfig -o /boot/grub/grub.cfg

You can install the same update-grub command on arch linux with this aur package: https://aur.archlinux.org/packages/update-grub/