$\newcommand\de\delta\newcommand{\KL}{\operatorname{KL}}$The maps $$\mu\mapsto\sqrt{\KL(\mu\parallel\nu)}$$ and $$\nu\mapsto\sqrt{\KL(\mu\parallel\nu)}$$ are not convex in general.
Indeed, let $\mu_p:=p\de_0+(1-p)\de_1$, where $p\in(0,1)$ and $\de_a$ is the Dirac measure supported on $\{a\}$.
Then the second partial derivative with respect to $p$ of $\sqrt{\mathrm{KL}(\mu_p,\mu_r)}$ at $(p,r)=(1/10,1/11)$ is $-7.17\ldots<0$. So, $\sqrt{\mathrm{KL}(\mu,\mu_r)}$ is not convex in $\mu$.
Also, the second partial derivative with respect to $r$ of $\sqrt{\mathrm{KL}(\mu_p,\mu_r)}$ at $(p,r)=(1/10,1/9)$ is $-11.50\ldots<0$. So, $\sqrt{\mathrm{KL}(\mu_p,\nu)}$ is not convex in $\nu$.
You also asked: "If this is not true in general, does it exists a measure $\mu\in P(X)$ such that $\nu\mapsto \sqrt{\mathrm{KL}(\mu\parallel\nu)}$ is a convex map $P(X)\to \mathbb [0,+\infty]$? Or a measure $\nu$ such that $\mu\mapsto \sqrt{\mathrm{KL}(\mu\parallel\nu)}$ is convex?"
The answer to each of these two questions is yes, at least when $X=\{0,1\}$, say.
For $p\in(0,1)$, let
\begin{equation}
F(p):=\sqrt{\mathrm{KL}(\mu_p,\mu_{1/2})},
\end{equation}
\begin{equation}
f(p):=F''(p)4 \mathrm{KL}(\mu_p,\mu_{1/2})^{3/2},
\end{equation}
\begin{equation}
f_1(p):=f'(p)(1-p)^2 p^2.
\end{equation}
Then $f_1(1/2)=f'_1(1/2)=f''_1(1/2)=0$ and
\begin{equation}
f'''_1(p)=\frac{2+4 p(1-p)}{(1-p)^2 p^2}>0.
\end{equation}
It follows that $F''(p)\ge0$, so that $\sqrt{\mathrm{KL}(\mu,\mu_{1/2})}$ is convex in $\mu$.
For $r\in(0,1)$, let
\begin{equation}
G(r):=\sqrt{\mathrm{KL}(\mu_{1/2},\mu_r)},
\end{equation}
and
\begin{equation}
g(r):=G''(r)4\mathrm{KL}(\mu_{1/2},\mu_r)^{3/2}.
\end{equation}
Then $g(1/2)=g'(1/2)=g''(1/2)=g'''(1/2)=0$ and
\begin{equation}
g''''(1/2+h)\frac{(1 - 4 h^2)^4}{16}=9- 16 h^4 + 156 h^2 + 64 h^6
>9- 1 + 156 h^2 + 64 h^6>0
\end{equation}
if $|h|<1/2$.
It follows that $G''(r)\ge0$, so that $\sqrt{\KL(\mu_{1/2},\nu)}$ is convex in $\nu$.
Remark: The existence of a probability measure $\nu\in P(X)$ such that $\sqrt{\KL(\mu\parallel\nu)}$ is convex in $\nu$ holds for any Polish space $X$. Indeed, the case when $X$ is a singleton set is trivial. Otherwise, take any distinct points $x$ and $y$ in $X$ and let $\nu:=\frac12\,\de_x+\frac12\,\de_y$. Then the condition $\mu\ll\nu$ implies $\mu=p\,\de_x+(1-p)\,\de_y$ for some $p\in[0,1]$, so that here $\sqrt{\KL(\mu,\nu)}=F(p)$.
