Not unless you have direct access to EK's res system. Sometimes, agents will tell you when you check in, sometimes staff will let you know (at EK ticket offices) but it is rare.

The best source of info (which is not 100% accurate) is Expert Flyer. I could always tell if a flight was overbooked based on what fare classes were still available (or not). If a flight is showing all zeros for example in economy or business, you can confirm this by calling EK and asking some innocent questions.

Remember, the load numbers are dynamic and are constantly moving up/down.

Is there a particular flight you are wanting info on?