Pokemon Legends: Z-A is the first core series game to not feature a method of traversing water using Pokemon as transport at any point in the base game. Your trainer cannot surf at all and invisible walls extending high into the sky will block the player character from passing over deep water. Surf can be used as a field move to clear away the piles of purple sludge littered across the city, but it's not particularly special since other Water-type moves also work.
Some areas that are covered by a shallow layer of water are still walkable, so check if there are any spots where there is actually solid ground just beneath the water. There are also locations where you can attack overworld objects to form bridges. As a notable example, the Rogue Mega Barbaracle mission is reached by walking into a tunnel filled with a shallow water and attacking an ice formation that forms an ice bridge across a gap of deep water.
Interestingly, leaked beta content reveals that rideable Pokemon were tested in early development, suggesting a similar system to the Ride Pokemon from Alola and Hisui (source, contains leaks for upcoming content), so it's possible that surfing was a planned but scrapped mechanic. It is currently unknown whether there are plans to add a surfing ability in future content for the game.
