 Solution

In the upper-half-plane model, with metric $d{s}^2=(d{x}^2+d{y}^2)/y^2$, 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 $s\mapsto 2s_i-s$: the perpendicular intersection point is fixed and the tangent direction along $h$ is reversed. Thus their composition restricts toThe 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 $N>0$. The composite is a hyperbolic translation of infinite order. The assumption excluding a common endpoint ensures the common perpendicular used here exists.