  We investigate the existence of minisuperspace description for the
homogeneous cosmological field equations within the framework of symmetric
teleparallel $f(Q)$-gravity. We consider the background space to be described
by the isotropic Friedmann--Lema\^{\i}tre--Robertson--Walker geometry, the
anisotropic Kantowski-Sachs and the anisotropic Bianchi III geometries. Across
all these models, we establish that the field equations in $f(Q)$-cosmology
exhibit second-order characteristics in the coincident gauge and those of a
sixth-order theory in the non-coincident gauge. Specifically, within the latter
scenario, the dynamic degrees of freedom are attributed to two scalar fields.
Finally, as an example of integrability, we derive a vacuum cosmological
solution within the non-coincident gauge.
