Remove and discard skin and bones from trout, if necessary. Flake trout into small pieces.
Pulse cream cheese and and next 8 ingredients in a food processor 7-8 times or until combined. Stir in flaked trout pieces. Transfer to a serving dish. Serve immediately or cover and chill up to 2 days. Let stand at room temperature for 30 minutes before serving with celery sticks or flatbread crackers.