Combine blackberries, ketchup, vinegar, brown sugar, Worcestershire, smoked paprika, and cayenne in a small saucepan over medium heat. Bring to a simmer and cook for 8 to 10 minutes, mashing the berries with a spoon, until the mixture thickens slightly and the berries break down completely.

Press the mixture through a fine-mesh strainer into a bowl, scraping with a spatula to extract as much liquid as possible. Discard the seeds. You should have about 3/4 cup smooth sauce. Set aside to cool slightly.

Position a rack in the center of the oven and preheat to 425 degrees F. Line a rimmed baking sheet with parchment paper. Pat the salmon fillets dry with paper towels, then brush with olive oil and season with 1/2 teaspoon salt and the black pepper.

Arrange the salmon skin-side down on the prepared sheet. Spoon about 1 tablespoon of the blackberry barbecue sauce over each fillet, spreading to coat the top. Reserve the remaining sauce for serving.

Bake for 12 to 15 minutes, depending on thickness, until the salmon flakes easily with a fork and the internal temperature reaches 125 to 130 degrees F for medium. The glaze should look set and slightly caramelized at the edges.

Transfer the salmon to plates. Spoon the reserved blackberry barbecue sauce alongside or over the top. Serve immediately.