Not very elegant, but you can combine two or three plots:

Show[Plot[{x^2, 2 x + 3}, {x, -3, 4}],
 Plot[{x^2, 2 x + 3}, {x, -1, 3}, Filling -> {1 -> {2}}]]

For even more control, use a custom ColorFunction in the 'bottom' plot that has the filling:

Show[Plot[{x^2, 2 x + 3}, {x, -3, 4}, Filling -> {1 -> {2}}, 
  ColorFunction -> 
   Function[{x, y}, 
    If[-1 < x < 3, RGBColor[0.817029, 0.811551, 0.899535], 
     RGBColor[1, 1, 1]]], ColorFunctionScaling -> False], 
 Plot[{x^2, 2 x + 3}, {x, -3, 4}]]

By changing the parameters if the If statement, you can make the filling be exactly where you want it.