In a way, a good trainer is that for a pokémon. While not every trainer is capable of doing this, it’s not unusual for a trainer to adopt and help pokémon with mental health needs of one sort or another. Moreover, pokémon centers often have pokémon psychologists—either Nurse Joy or another trained professional—on staff to offer advice and therapy.
In short, anonymous, no, there are no mental hospitals for pokémon. That’s generally more of a human concept, to be honest.
The Pokédex According to Bill 