Usually you will have a schematic in front of you when repairing an unknown piece of kit, and hence it would be obvious which components were FETs and which were transistors.
If fault finding without the luxury of a schematic (a technique I tend to avoid like the plauge) then your only remaining option is to plug all of the device numbers from within the area you are interested into google and do a mass datasheet download.
In practice I have found it very difficult to determine that an FET is likely to be good by cold checks alone. It's not the same as testing a transistor where you can measure the base-emitter junction for example.
Unless the FET in question has an obvious fault symptom, such as being completely short-circuit, the only sure way to test it is to remove the device and test it on some prototyping board under working conditions. Personally, if I suspected an FET and I had a replacement to hand I'd simply exchange it and then test the monitor again. If it works, you guessed correctly. If you still have the same fault symptom, you were wrong!
Brian