The statement
Let $X\sim\mathcal N(\mu,\sigma^2)$ denote a normally distributed random variable with mean $\mu$ and variance $\sigma^2$.
implies that $X$ admits a density and distribution function of the form
$$
f_X(x)=\frac{1}{\sqrt{2\pi}\sigma}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)
$$
and
$$
F_X(x)={\displaystyle {\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {x-\mu }{\sigma {\sqrt {2}}}}\right)\right]},
$$
respectively. There is no need to include an $x$ in the manner you specified, i.e. $\mathcal N_x(\mu,\sigma^2)$.
Note the use of the subscript $X$ in the notation pertaining the density and distribution functions to make it clear that they correspond to the random variable $X$. This is standard notation which can be used for other types of random variables. For example, $X\sim\operatorname{Gamma}(\alpha,\beta)$ denotes a gamma random variable and we can write the corresponding density as
$$
f_X(x)=\frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}.
$$