Let 
\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][1], 
\begin{equation}
	V_n:=\sqrt2\,(\sqrt Y_n-\sqrt{n-2})\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-2})  \\ 
	&\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}
\tag{3}\label{3}
\end{equation*}
if $c$ varies with $n$ so that $c-u\sqrt{n-2}$ 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.  

---

In fact, we have a lower bound on $P(x\in C)$ that is 
essentially the same as the limit in \eqref{3}:   
\begin{equation*}
\begin{aligned}
	P(x\in C)
	&=P(Z_1+\tfrac u{\sqrt2}\,V_n\ge c-u\sqrt{n-2})  \\ 
	&>P(Z_1+\tfrac u{\sqrt2}\,V\ge c-u\sqrt{n-2})  \\ 
	&=1-\Phi\Big(\frac{c-u\sqrt{n-2}}{\sqrt{1+u^2/2}}\Big).   
\end{aligned}
\tag{4}\label{4}
\end{equation*}
The inequality in \eqref{4} follows by [formula (2.6)][2], which means that $V_n$ is strictly stochastically greater than $V$. 

[1]: https://en.wikipedia.org/wiki/Delta_method#Univariate_delta_method 
[2]: https://projecteuclid.org/journals/annals-of-statistics/volume-22/issue-1/Extremal-Probabilistic-Problems-and-Hotellings-T2-Test-Under-a-Symmetry/10.1214/aos/1176325373.full