Let 
\begin{equation}
	Z=(Z_1,\dots,Z_n):=x/\sigma\sim N(0,I_n),
\end{equation}
\begin{equation}
	Y:=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}) \\
	&=1-P(Z_1<c-u\sqrt Y). 
\end{aligned}
\end{equation}
Note that the random variables $Z_1$ and $Y$ 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, 
\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.