Even if there isn't enough heat and energy in the star to cause nucleosynthesis, could atoms quantum entangle to create a new atomic nuclei? Or would it be quantum tunneling - if this is possible?
You are referring to pyconuclear reactions - these are reactions that are initiated even when the temperature is effectively zero. They are caused by zeropoint oscillations of particles that are trapped in a deep potential well and hence are purely a quantum mechanical effect.
This is far from just of theoretical interest. It may be that pyconuclear reactions occur in the condensed matter of white dwarfs; possibly even initiating the carbon burning that can destroy the star in a type Ia supernova.
I think you're referring to quantum tunneling. Well, quantum tunneling is actually essential in nucleosynthesis in stars. Usually quantum tunneling is a very rare occurence — but in a body as large as the Sun (whose diameter is 4 light seconds long), it is bound to happen quite frequently. This is what mainly allows atomic nuclei to collide.