\begin{equation*} Z=(Z_1,\dots,Z_n):=x/\sigma\sim N(0,I_n), \end{equation*} \begin{equation*} Y_n:=Z_2^2+\dots+Z_n^2\sim\chi^2_{n-1}, \end{equation*} \begin{equation*} c:=\|v\|_2/\sigma>0,\quad t:=\cos\theta\in(0,1),\quad u:=\frac t{\sqrt{1-t^2}}=\cot\theta\in(0,\infty). \end{equation*} By the spherical symmetry, without loss of generality $v=c\sigma(1,0,\dots,0)$. So, \begin{equation*} \begin{aligned} P(x\in C)&=1-P(c-Z_1>t\sqrt{(Z_1-c)^2+Y_n}) \\ &=1-P(Z_1<c-u\sqrt Y_n). \end{aligned} \tag{1}\label{1} \end{equation*} Note that the random variables (r.v.'s) $Z_1$ and $Y_n$ are independent. So, \begin{equation*} P(x\in C)=1 -\frac{2^{(1-n)/2}} {\Gamma ((n-1)/2)} \int_0^\infty \Phi(c-u\sqrt y) e^{-y/2} y^{(n-3)/2}\,dy, \tag{2}\label{2} \end{equation*} where $\Phi$ is the standard normal cdf.
Mathematica cannot do anything with the latter integral. So, it is unlikely that it can be expressed in closed form.
However, using \eqref{1} or \eqref{2}, one can easily find various approximations to $P(x\in C)$, depending on how $n,c,u$ vary.
For instance, suppose that $u$ is fixed and $n\to\infty$. Then, by the central limit theorem and the delta method, \begin{equation} V_n:=\sqrt2\,(\sqrt Y_n-\sqrt n)\to V \end{equation} in distribution, where $V$ is a standard normal r.v., which let us choose to be independent of $Z_1$. Then, by \eqref{1}, \begin{equation*} \begin{aligned} P(x\in C)&=P(Z_1\ge c-u\sqrt Y_n) \\ &=P(Z_1+\tfrac u{\sqrt2}\,V_n\ge c-u\sqrt n) \\ &\to P(Z_1+\tfrac u{\sqrt2}\,V\ge c_0) \\ &=1-\Phi\Big(\frac{c_0}{\sqrt{1+u^2/2}}\Big) \end{aligned} \end{equation*} if $c$ varies with $n$ so that $c-u\sqrt n$ converges to some real $c_0$.
Similarly, if $u$ and $c$ are fixed whereas $n\to\infty$, then $P(x\in C)\to1$.
Of course, one can also use various asymptotic expansions to obtain more detailed asymptotics.