The hexacyclic system graph \( F_n \) is the graph derived from a linear hexagonal chain \( L_n \) with \( n > 1 \) hexagons by identifying two pairs of ends of \( L_n \). The Möbious hexacyclic system graph \( M_n \) is the graph derived from a linear hexagonal chain \( L_n \) with \( n > 1 \) hexagons by identifying two pairs of ends of \( L_n \) with a twist. In this paper, we compute, in a closed form, the resolvent energy, the Laplacian and the signless Laplacian resolvent energy, as well as the resolvent Estrada index and the resolvent signless Estrada index of \( F_n \) and \( M_n \). All five indices are expressed as a rational function in the number \( n \) of hexagons, defined in terms of Chebyshev polynomials of the first and the second kind. Those expressions allow for a fast numerical computation of indices and for deducing sharp bounds on their growth.