Can a Mac OS X 10.6.8 be used as a screen for apple4k tv

You may be able to use your TV as a Mac screen,

but you can’t use your Mac as a screen for the Apple TV box.

The Apple TV box needs to connect to a TV, monitor, or projector that has HDMI-in. Computers have HDMI-out (if they have HDMI). The ports are directional.

Highly unlikely, but we need more. What Mac? From "About this Mac..."

Click the button I've marked with the blue arrow and, on the resulting window:

tell us the code for your Mac where I've highlighted with the red arrow.