A search for flavour-changing neutral current (FCNC) $tqH$ interactions involving a top quark, an up-type quark ($q=u, c$), and a Standard Model (SM) Higgs boson decaying into a $\tau$-lepton pair ($H\rightarrow \tau^+\tau^-$) is presented. The search is based on a dataset of $pp$ collisions at \sqrt{s}=13~\tev recorded with the ATLAS detector at the Large Hadron Collider that corresponds to an integrated luminosity of 139 fb$^{-1}$. Two processes are considered: single top quark FCNC production in association with a Higgs boson ($pp\rightarrow tH$), 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 $\tau$-lepton candidates (\tauhad) or at least one \tauhad with an additional lepton (e,$\mu$), as well as multiple jets. Event kinematics are used to separate signal from the background through a multivariate discriminant. A slight excess of data is observed above the expected SM background with a significance of 2.3$\sigma$, and 95\% CL upper limits on the $t\to qH$ branching ratios are derived. Observed (expected) 95\% CL upper limits are set on the $t\to cH$ and $t\to uH$ branching ratios of $9.9 \times 10^{-4}$ ($5.0^{+2.2}_{-1.4}\times 10^{-4}$) and $7.2\times 10^{-4}$ ($3.6^{+1.7}_{-1.0}\times 10^{-4}$), respectively. The corresponding combined observed (expected) upper limits on the dimension-6 operator Wilson coefficients in the effective $tqH$ couplings are C_{c\phi} <1.38\, (0.97) and C_{u\phi} <1.18\, (0.83), respectively.

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