Combine cucumbers, onions, red bell pepper, and green bell pepper in a large bowl; add salt and pour in enough cold water to cover. Allow to soak for 2 hours. Thoroughly drain in a colander.
Pour vinegar into a large, heavy stainless steel pot; add celery seed and mustard seed. Stir sugar into mixture and bring to a boil, stirring until sugar is dissolved, 2 to 3 minutes. Add the drained vegetable mixture and return to a boil, about 10 minutes. Remove pot from heat.
Sterilize the jars and lids in boiling water for at least 5 minutes. Pack relish into hot, sterilized jars, filling to within 1/4 inch of the top. Run a clean knife or thin spatula around the insides of the jars after they have been filled to remove any air bubbles. Wipe the rims of the jars with a moist paper towel to remove any food residue. Top with lids and screw on rings.
Place a rack in the bottom of a large stockpot and fill halfway with water. Bring to a boil and lower jars into the boiling water using a holder. Leave a 2-inch space between the jars. Pour in more boiling water if necessary to bring the water level to at least 1 inch above the tops of the jars. Bring the water to a rolling boil, cover the pot, and process for 10 minutes.
Remove the jars from the stockpot and place onto a cloth-covered or wood surface, several inches apart, until cool. Press the top of each lid with a finger, ensuring that the seal is tight (lid does not move up or down at all). Store in a cool, dark area.