Like several European countries, Switzerland has no single nationwide pet-emergency phone number. Instead, regular vet clinics run or refer to an on-call rota (Notfalldienst in German, service d'urgence in French, servizio d'urgenza in Italian) that covers evenings, nights and weekends within a region.
The most reliable way to find help quickly is to ask your accommodation, or search the term above plus your city, before you actually need it — save two or three numbers in your phone on arrival, the way you would with any trip.
Switzerland's veterinary care is genuinely excellent by any European standard, though it comes at a price: a standard consultation typically runs CHF 60–120, with emergency visits costing more. Pet insurance is worth considering for a longer trip.