In a food processor, combine cream cheese, red peppers, and lemon juice. Process until well combined.
Using a mandoline or a sharp knife, very thinly slice cucumber crosswise into rounds.
Place 1 slice white bread and 1 slice wheat bread on work surface. Spread one side of each slice with a thin layer of the cream cheese. Top white bread with a thin layer of cucumber slices; place wheat bread cream cheese-side down over cucumber slices. Press together firmly. Repeat with remaining ingredients.
Chill sandwiches, covered, until firm. Using a sharp knife, remove crusts. Cut sandwiches into quarters before serving.