APU may be used in flight as second source of power or bleed air if one of the engine systems is inoperative. If it is so used, it might leave a contrail the same way engines will, though as DeltaLima notes, it's exhaust is hotter and the air behind the fuselage is more turbulent, so the range of conditions where it will leave a contrail is narrower than for the engines.
And it is an exceptional situation, so it is extremely rare. Most of the time the APU is switched off once the engines are running and only switched on again after landing, because when engines are running, they provide everything the aircraft needs.
In B737NG (the lower aircraft in the picture), the APU can be used as bleed air source up to 17,000 ft. This is too low for normal cruise, but if bleed system fails, pilots will descend to this altitude and start APU to provide air-conditioning for the remainder of flight to a suitable nearby airport. It can also be used up to 41,000 ft as power source, so the aircraft may cruise with one generator inoperative and running APU.
In A320 (the upper aircraft in the picture), the APU can be used together with one bleed air system up to 20,000 ft and for power generation up to 41,000 ft. So again, the aircraft may cruise with one generator inoperative and running APU.
- Normally there would never be contrail behind APU and none behind engines unless both engines failed and APU was started as backup power source (US1549 did start APU when the engines failed, though it is unlikely there was contrail at that altitude).
- Contrail does not start right behind the engines. The water in the exhaust needs some time to cool down and freeze before the contrail becomes visible.