Once it's clear that any two points with multiple connecting geodesics are conjugate points, it's a straightforward application of the Jacobi equation to get that there are no conjugate points, as seen in Why do manifolds with negative sectional curvature not have conjugate points?. However, I wasn't sure I had a correct inclusion of simply connected. My argument is as follows:
Take two geodesics connecting $p$ and $q$, say $\exp_p(tv)$ and $\exp_p(tw)$, with $v\not=cw$ for any positive $c$ and $q=\exp_p(v)=\exp_p(w)$. Then $\gamma_s(t)=\exp_p(t*(v+s*(w-v))$ is a single-parameter family of geodesics, and $w-v\in\ker(d(\exp_p)_v)$, hence $s(w-v)\in\ker(d(\exp_p)_v)$, so $\gamma_s(1)=$q for all $s$.
Simply connected should come into play in ensuring that changing $s$ is a smooth transition from one geodesic to the other, but I wasn't sure if it was enough to just say that or if there was more to be proven.