Probing the radial collective oscillation of a trapped quantum system is an accurate experimental tool to investigate interactions and dimensionality effects. We consider a fully polarized quasi-one-dimensional dipolar quantum gas of bosonic dysprosium atoms in a parabolic trap at zero temperature. We model the dipolar gas with an effective quasi-one-dimensional Hamiltonian in the single-mode approximation and derive the equation of state using a variational approximation based on the Lieb-Liniger gas Bethe ansatz wave function or perturbation theory. We calculate the breathing mode frequencies while varying polarization angles by a sum-rule approach and find they are in good agreement with recent experimental findings.