No, impulse only works in space. In atmosphere impulse would likely rip the Enterprise itself to pieces because it can move the Enterprise to like 1/10th the speed of light or something. In atmosphere the Enterprise would move on thrusters which is used for moving the ship into or out of orbit, evasive maneuvers, etc. while impulse would be used for general space travels. Warp drive does not actually move the ship faster than the speed of light but instead creates a field in which the ship would appear to move faster than light without turning a 1 hour trip into a 100,000,000 year trip (from the point of view of the rest of the Federation). The plane would probably not survive the warp transition however the Enterprise can always extend its shield around it to protect it. Now the Enterprise (the original Constitution class Enterprise) is about 1000 feet long, so a 747 would look small compared to the Enterprise.
So in short, the Enterprise can carry the plane even if the plane can't survive the rigors of space travel by extending its shield around it. Heck the plane could probably fit inside the ship.... (maybe the Enterprise-E)