A search for flavour-changing neutral current (FCNC) tqH interactions involving a top quark, another up-type quark (q = u, c), and a Standard Model (SM) Higgs boson decaying into a τ-lepton pair (H → τ$^{+}$τ$^{−}$) is presented. The search is based on a dataset of pp collisions at $ \sqrt{s} $ = 13 TeV that corresponds to an integrated luminosity of 139 fb$^{−1}$ recorded with the ATLAS detector at the Large Hadron Collider. Two processes are considered: single top quark FCNC production in association with a Higgs boson (pp → tH), and top quark pair production in which one of top quarks decays into Wb and the other decays into qH through the FCNC interactions. The search selects events with two hadronically decaying τ-lepton candidates (τ$_{had}$) or at least one τ$_{had}$ with an additional lepton (e, μ), as well as multiple jets. Event kinematics is used to separate signal from the background through a multivariate discriminant. A slight excess of data is observed with a significance of 2.3σ above the expected SM background, and 95% CL upper limits on the t → qH branching ratios are derived. The observed (expected) 95% CL upper limits set on the t → cH and t → uH branching ratios are $ 9.4\times {10}^{-4}\left({4.8}_{-1.4}^{+2.2}\times {10}^{-4}\right) $ and $ 6.9\times {10}^{-4}\left({3.5}_{-1.0}^{+1.5}\times {10}^{-4}\right) $, respectively. The corresponding combined observed (expected) upper limits on the dimension-6 operator Wilson coefficients in the effective tqH couplings are C$_{cϕ}$< 1.35 (0.97) and C$_{uϕ}$< 1.16 (0.82).