Drain the capers and finely chop them.
In a small bowl or glass measuring cup, whisk together the chopped capers, mayonnaise, 2 tablespoons lemon juice, and 1 tablespoon dill pickle relish until smooth.
Season the caper mayonnaise with salt and pepper to taste, then refrigerate while you prepare the salmon burgers.
Coarsely chop celery, onion, and fresh chives or green onion and add them to a food processor along with the other burger mix-ins called for in your main recipe (hemp hearts, almond flour, egg, sriracha, lemon juice, salt, and pepper).
Pulse the mixture just until the vegetables are chopped and the ingredients are combined.
Add the diced salmon cubes to the food processor and pulse a few more times so the salmon is barely chopped and evenly mixed with the other ingredients.
With wet hands, form the mixture into six medium burgers and place them on a baking sheet; optionally chill in the freezer about 20 minutes to help them hold together.
Preheat the grill to medium-high and spray the grates with non-stick spray. Brush both sides of each burger lightly with olive or grapeseed oil.
Grill the burgers without moving them for about 4 minutes, then check for grill marks. If desired, rotate each burger for criss-cross marks and cook 2–3 minutes more on that side.
Flip the burgers and grill 2–4 minutes more on the second side, until nicely browned and firm to the touch, or until an instant-read thermometer registers 145°F (62°C).
Serve the burgers hot with the chilled caper mayonnaise, on a bun with lettuce and tomato, or on a plate as desired.