We study the integrable structure and scaling limits of the conditioned eigenvector overlap of the symplectic Ginibre ensemble of Gaussian non-Hermitian random matrices with independent quaternion elements. The average of the overlap matrix elements constructed from left and right eigenvectors, conditioned to $x$, are derived in terms of a Pfaffian determinant. Regarded as a two-dimensional Coulomb gas with the Neumann boundary condition along the real axis, it contains a kernel of skew-orthogonal polynomials with respect to the weight function $ω^{(\mathrm{over})}(z)=|z-\overline{x}|^2(1+|z-x|^2)e^{-2|z|^2}$, including a non-trivial insertion of a point charge. The mean off-diagonal overlap is related to the diagonal (self-)overlap by a transposition, in analogy to the complex Ginibre ensemble. For $x$ conditioned to the real line, extending previous results at $x=0$, we determine the skew-orthogonal polynomials and their skew-kernel with respect to $ω^{(\mathrm{over})}(z)$. This is done in two steps and involves a Christoffel perturbation of the weight $ω^{(\mathrm{over})}(z)=|z-\overline{x}|^2ω^{(\mathrm{pre})}(z)$, by computing first the corresponding quantities for the unperturbed weight $ω^{(\mathrm{pre})}(z)$. Its kernel is shown to satisfy a differential equation at finite matrix size $N$. This allows us to take different large-$N$ limits, where we distinguish bulk and edge regime along the real axis. The limiting mean diagonal overlaps and corresponding eigenvalue correlation functions of the point processes with respect to $ω^{(\mathrm{over})}(z)$ are determined. We also examine the effect on the planar orthogonal polynomials when changing the variance in $ω^{(\mathrm{pre})}(z)$, as this appears in the eigenvector statistics of the complex Ginibre ensemble.