This is the Black Scholes Call Price:
\begin{align}
C(S, t) &= N(d_1)S - N(d_2) Ke^{-r(T - t)} \\
d_1 &= \frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{S}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)(T - t)\right] \\
d_2 &= \frac{1}{\sigma\sqrt{T - t}}\left[\ln\left(\frac{S}{K}\right) + \left(r - \frac{\sigma^2}{2}\right)(T - t)\right] \\
&= d_1 - \sigma\sqrt{T - t}
\end{align}
All parameters except the underlying price $S$ are assumed constant. $S$ has a lognormal distribution and follows a GBM under $Q$:
$$S_t=S_0e^{(r-\frac{\sigma^2}{2})t+\sigma W_{t}^{Q}}$$
You can directly observe from the $C(S,t)$ formula that the distribution of $C$ cannot be in closed form since $N(*)$ is not in closed form.
You can simulate the distribution of $C$ by drawing many samples from $W\sim N(0,T)$.