Short answer: possibly, but not proven. Psilocybin converts to psilocin, which in cell models can modulate immune signaling through 5‑HT2A receptors. That could indirectly affect how tryptophan is used.
Under stress and inflammation, IDO and TDO enzymes divert tryptophan into the kynurenine pathway, sometimes called a tryptophan steal. If psilocin reduces pro‑inflammatory cytokines, IDO/TDO activity may drop and relatively more tryptophan could remain available for serotonin synthesis.
Evidence is limited. In vitro studies report dose and context dependent reductions in markers like TNF‑alpha and IL‑6, but small studies in healthy volunteers have not shown clear changes in peripheral inflammatory markers after a single dose. There is no direct human evidence yet that psilocybin or psilocin increases bioavailable tryptophan or shifts the kynurenine to tryptophan ratio.
So the mechanism is biochemically plausible, especially in inflamed states, but I cannot verify it in humans at this time. For background on psilocybin, see this overview. Hope this helps.