Furfrou can appear in different in forms in Pokemon Legends: Z-A, which all depend on the possible trim you get for it, but the Furfrou Trimmer isn’t available immediately. From what I’ve seen, you first need to unlock the Wild Zone that contains the Pokemon. Then, once you do, a side mission will appear nearby it that unlocks the feature.
Where is the Furfrou Salon in Pokemon Legends: Z-A?
You won’t be able to even see an option for the Furfrou Salon in the game until side mission 47 appears on the Lumiose City map. From what I found, this doesn’t open up until Wild Zone 11 is available. This happens as you progress the story.
Once the mission is available, you’ll find it in this section of Jaune Sector 6, not far from Justice Dojo and the Northwest most corner of Wild Zone 11.

A woman named Mirte will be standing there with her Scyther. You need to have a Pokemon that knows the move Slash use that attack on the Substitute doll near her so Scyther can learn the move for trimming Furfrous. (If you picked Totodile, it knows it.)
After you do, Mirte and her Furfrou Salon will open for business and trims in Pokemon Legends: Z-A. Unfortunately, this won’t add an icon to the map showing where the Furfrou trimmer is for new trim forms at all times in Pokemon Legends: Z-A, even after you use her services. It will cost 500 for a cut, with Heart, Star, and Diamond initially available. Additional form options are unlocked via subsequent side missions like Side Mission 62: Becoming a Pro Furfrou Trimmer. Have a Pokemon with Razor Leaf on hand for that one!
Pokemon Legends: Z-A is available for the Switch and Switch 2.