Usually only to a limited extent. In-camera JPEG creation is often handled partly by specialized image-processing hardware, not just easily changed software. Because of that, firmware updates may tweak things like processing parameters or behavior, but you generally should not expect a major transformation in JPEG quality from firmware alone.

So while small improvements are possible, a significant overhaul of sharpening, white balance, or artifact handling is less likely if the underlying processing pipeline is constrained by the camera’s hardware.