A search for flavour-changing neutral interactions involving the top quark, the Higgs boson and an up-type quark q (q = c, u) is presented. The proton-proton collision data set used, with an integrated luminosity of 139 fb$^{−1}$, was collected at $ \sqrt{s} $ = 13 TeV by the ATLAS experiment at the Large Hadron Collider. Both the decay process t → qH in $ t\overline{t} $ production and the production process pp → tH, with the Higgs boson decaying into two photons, are investigated. No significant excess is observed and upper limits are set on the t → cH and the t → uH branching ratios of 4.3 × 10$^{−4}$ and 3.8 × 10$^{−4}$, respectively, at the 95% confidence level, while the expected limits in the absence of signal are 4.7 × 10$^{−4}$ and 3.9 × 10$^{−4}$. Combining this search with ATLAS searches in the H → τ$^{+}$τ$^{−}$ and H → $ b\overline{b} $ final states yields observed (expected) upper limits on the t → cH branching ratio of 5.8 × 10$^{−4}$ (3.0 × 10$^{−4}$) at the 95% confidence level. The corresponding observed (expected) upper limit on the t → uH branching ratio is 4.0 × 10$^{−4}$ (2.4 × 10$^{−4}$).