Put simply, no.
Put less simply, perhaps (if you're willing to economise a little).
To be more specific; a lot has been made in the past about various early cultures all building pyramidal monumental structures, especially ziggurats - Egypt, Sumeria, Mesopotamia, Babylon, just to name a few. These were not all built to similar designs because of ancient alien influence...
Alright, I'll admit I don't know that for sure...
but because the design provides stability when building with the materials available to those societies at the time. Put simply, the angle of incline meant you could build as high as possible without the structure collapsing. It wasn't until we developed engineering concepts like arches that we see other monumental structures and grand buildings like triumphal arches and even the Colosseum.
The thing is, arches are designed to distribute weight from a large section of 'roof' to a small section of pillared support. This works to keep up most buildings, sure. But mountains, more specifically Everest, weigh a lot more than a stone building. Even if arches could support the weight, you'd need so many of them that your grand hall and court for your emperor would have far too many readily available hiding spots in it for your guardsmen's liking simply because of the plentitude of arches needed to support the rest of the mountain.
Add to that, as you pointed out in your question, the mountain is subject to tectonic activity and has basically been created by the collision of two major tectonic plates. In short, that means that the mountain is moving (albeit ever so slowly) and actually growing over time.
This becomes a problem for your hollowing activities, because the focal points on your cavern roofs for the support columns and arches are actually going to shift a little over time, meaning that your existing columns are likely to shift and potentially break or tip over at some point in the future, leaving your now hollow mountain less stable, potentially accelerating the process.
Structurally, if your emperor wanted to do this today, he probably could but it would cost a lot. You'd need pillars built or braced in a manner that allows movement, and you'd also need to know the general direction of the tectonic forces within the mountain in order to design the hollowed out chambers to be supported in a direction and manner that provided longevity by addressing the direction of change within the chamber once it is hollowed out.
In short, the hollowing out of Everest wouldn't increase tectonic activity, but it would destabilise the mountain itself, which from an inhabitant's perspective would have very similar effects to an acceleration of tectonic activity.
Of course, this all assumes that you're hollowing out a significant percentage of the mountain. Based on one estimate I found on the internet, the volume of Everest is around 1.4 trillion M3, which is a lot. To put that in perspective, if the total floor area of the Empire State Building is estimated at 210k m2 and we presume a very generous 5m ceiling, we can safely round this down a little to 1 million m3</sup), which is less than 0.0001% of the mountain. That could be done relatively safely, especially if you modularise your caverns and distribute them through the mountain relatively evenly and you still have a palace larger than the Empire State building in terms of floor space, which (let's face it) would be a very grand place to live assuming all you cared about is size.
I'll nail my colours to the mast right now and say that it's probably not wise to do this, anymore than it would be wise to build any house on a fault line. But, if you're really just interested in whether or not it's possible, then the above gives you some context about how it might be done.
