As a company, no. Pokémon is an international company, and that includes countries like China where the concept of LGBTQ+ is still very much against societal values. Running a pride month event would likely face massive backlash and lower profits.
However, TPC has admitted to supporting the LGBTQIA+ community before. A quick Google search reveals this incident, where TPC representative openly criticizes Chinese Pokémon Unite developer TiMi studio's tweet and responds by stating their support for the community. Moreover, Pokémon Go developer Niantic and Nintendo themselves have both participated in pride month parades, as seen here, so it's clear that they support it when they can.
