= Solution
In the upper-half-plane model, with metric $ds^2=(dx^2+dy^2)/y^2$, the <hyperbolic geodesics> are vertical lines and semicircles <orthogonal> to the real boundary. Two <geodesics> with disjoint endpoints and no intersection are ultraparallel and have a common perpendicular $h$. To construct it, send one <geodesic> to the imaginary axis by an <isometry>. The second then has endpoints $0<a<b$ after a possible reflection. The semicircle centered at zero with radius $\sqrt{ab}$ is perpendicular to the imaginary axis and to the second <circle>: its squared radius plus $((b-a)/2)^2$ equals $((a+b)/2)^2$, the orthogonality condition for the two <circles>.
Let the intersection points of $h$ with the given <geodesics> have signed arclength coordinates $s_1,s_2$ on $h$. Each <hyperbolic reflection> preserves $h$ and restricts there to $s\mapsto2s_i-s$: the perpendicular intersection point is fixed and the tangent direction along $h$ is reversed. Thus their composition restricts to
$$
s\longmapsto2s_1-(2s_2-s)=s+2(s_1-s_2).
$$
The displacement is nonzero since the two perpendicular intersection points differ. Its $N$th iterate translates by $2N(s_1-s_2)$, which is never zero for $N>0$. \b[The composite is a hyperbolic translation of infinite order.] The assumption excluding a common endpoint ensures the common perpendicular used here exists.
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