$C$, $D$ are two categories. $F$, $G$ are functors between $C$ and $D$: $F, G: C\rightarrow D$. Let $Nat(F,G)$ be all the natural transformations between F and G.
Like what we do for groups, define the center of the a category $C$: $Z(C)=Nat(id_C,id_C)$.
Then what is the center of the fundamental groupoid $Z(\pi_{1}(X))$?
This is not really a question about centers of fundamental groupoids but centers of groupoids in general.
If $\newcommand{\G}{\mathcal{G}}\G$ is a group, considered as a 1-object category, then $Z(\G)$ (in the usual sense) is $\newcommand{\Nat}{\mathrm{Nat}}\newcommand{\id}{\mathrm{id}}\Nat(\id_\G,\id_\G)$. I claim that if $\G$ is a connected groupoid then $\Nat(\id_\G,\id_\G)$ is canonically isomorphic to $Z(\mathrm{Hom}_\G(x,x))$ for any choice of an object $x$ of $\G$. Indeed, pick an element $g \in Z(\mathrm{Hom}_\G(x,x))$. For any object $y$ of $\G$, define an element $g_y \in \mathrm{Hom}_\G(y,y)$ by choosing an arbitrary isomorphism $f: x \to y$ and setting $g_y = f \circ g \circ f^{-1}$. If we had chosen a different $f'$, then there would be an automorphism $h$ of $x$ such that $f' = f \circ h$. But $$ f \circ h \circ g \circ h^{-1} \circ f^{-1} = f \circ g \circ f^{-1},$$ by the assumption that $g$ is in the center, and therefore $g_y$ is well defined independent of the choice of $f$. One can now check that the elements $g_y$ define a natural transformation from $\id_\G$ to itself and that all natural transformations arise this way.
