The second (v ₂ ) and third (v ₃ ) flow harmonic coefficients of J/ψ mesons are measured at forward rapidity (2.5 < y < 4.0) in Pb-Pb collisions at sNN=5.02 TeV with the ALICE detector at the LHC. Results are obtained with the scalar product method and reported as a function of transverse momentum, p _T , for various collision centralities. A positive value of J/ψ v ₃ is observed with 3.7σ significance. The measurements, compared to those of prompt D ⁰ mesons and charged particles at mid-rapidity, indicate an ordering with v _n (J/ψ) < v _n (D ⁰ ) < v _n (h ^± ) (n = 2, 3) at low and intermediate p _T up to 6 GeV/c and a convergence with v ₂ (J/ψ) ≈ v ₂ (D ⁰ ) ≈ v ₂ (h ^± ) at high p _T above 6–8 GeV/c. In semi-central collisions (5–40% and 10–50% centrality intervals) at intermediate p _T between 2 and 6 GeV/c, the ratio v ₃ /v ₂ of J/ψ mesons is found to be significantly lower (4.6σ) with respect to that of charged particles. In addition, the comparison to the prompt D ⁰ -meson ratio in the same p _T interval suggests an ordering similar to that of the v ₂ and v ₃ coefficients. The J/ψ v ₂ coefficient is further studied using the Event Shape Engineering technique. The obtained results are found to be compatible with the expected variations of the eccentricity of the initial-state geometry.