Generally, if your camera has an HDMI output, then it will display the photos on the TV that it is plugged in to, but I'm not sure what you mean by USB HDMI cable? USB and HDMI are not the same, there is a technology called MHL that allows HDMI to be output over the same physical port as USB if it has special hardware, however the D600 doesn't have this, it just has a compact HDMI port. If you are trying to use an MHL adapter with the USB port, it is not going to work and may cause harm to the camera.
You also mention the HDMI connector on your laptop. Your laptop almost certainly has an HDMI output, it can't be used in both directions. You can run information out of the laptop to a TV, but you can't run HDMI from your camera in to the laptop. Attempting to do this may even damage either the camera or your laptop. If you plug the camera in to a TV via HDMI, then for most cameras, whatever would normally show up on the LCD should be displayed on the TV and the controls on the camera will function as normal.