gt.geometric topology – Quasi-isometric rigidity of surface groups and commensurability

Yes (I assume that by “virtually isomorphic” you mean commensurable modulo finite kernels, which is a nonstandard misleading use of “virtually”). This is because surface groups have Serre’s property (I forgot its name, maybe “P2” or so) meaning that each 2-cohomology class (in a finite abelian group with trivial action) it trivial on some finite index subgroup.

(Also a side remark: the known result on QI rigidity is stronger, since it says that every group QI to this surface group is, modulo a finite kernel, isomorphic to a cocompact lattice in the isometry group of the hyperbolic plane. I.e., there’s a structural statement which does not require passing to a finite index subgroup.)