Can you breed fish in Minecraft?

No.

You cannot breed Fish. There are no Baby Fish in Minecraft. Fish spawn naturally in Ocean Biomes, like passive mobs do on land.
_{- How do you breed fish? - Reddit}

The Wiki article also never mentions anything about breeding.

Fish can only spawn underwater at a Y level of 12-32.

Cod spawn in normal, cold, and lukewarm oceans, and their deep variants in groups of 4-7.

Pufferfish spawn in lukewarm, warm, and deep lukewarm oceans in groups of 3-5.

Tropical fish can only spawn in warm ocean in groups of 3-5.

Salmon spawn in groups of 3-5 in cold and frozen oceans, and their deep variants. Unlike the other types of fish, they also spawn in normal and frozen rivers. Salmon can spawn in 3 sizes which are small, normal and large.‌
_{- Fish (mob) - Minecraft Wiki}

So the ways listed above are the only ways fish can spawn.

Fish cannot breed. There is no mention of them having the ability to breed on the Official Minecraft Wiki, especially not in the spawning section.