How to model a noisy Zener diode in LTSPICE?
Spice models generally do not include noise in transient simulations. The "noise" model in spice is used only in AC sweeps, where the noise power is calculated as a function of frequency. While resistor (Johnson–Nyquist) noise is in the model, semiconductor models often do not have accurate noise models. The spice diode model does include flicker noise, but not other noise sources.

For your purposes, AC analysis may be sufficient assuming that your diode has a proper model, since what you want is to see if the noise power density is flat. But, I doubt that the Zener model includes accurate noise parameters. The spice model of the diode mentioned in this question (EDZV24B) does not include any noise parameters (which are the AF, KF, and FFE parameters).

Another option (for transient simulations) is to include a voltage sources controlled by a random number. For a description of using this approach to noise modeling, this website from Giorgio Vazzana has good information. But, to follow this approach, you have to know how much noise to expect. Also, the transient simulation would not normally include noise added by the transistor.

An example noise voltage source (from the above mentioned website) is:

Vn 1 n1 dc 0V ac 1mV trrandom (1 5us 0s 125m 0m)