France doesn't have one central pet-emergency phone number the way it does for human emergencies (15 or 112). Instead, most regular vet practices (cliniques vétérinaires) either run their own out-of-hours rota or refer after-hours calls to a dedicated regional emergency service, and that referral number is usually on the practice's own answering message.
If you're staying somewhere for more than a day or two, save a local vet or emergency number as soon as you arrive rather than searching for one mid-emergency. Searching "vétérinaire urgence" plus your city or département is the quickest way to find the regional on-call service if you're starting from nothing.
French veterinary care is generally well-regarded, and standard consultations are often more affordable than in the UK or Switzerland, though out-of-hours or emergency consultations cost noticeably more than a daytime appointment. Have a payment method ready for an unplanned, larger charge, since most practices expect payment at the time of treatment.