There is the SAGA tool "Raster values to points" (in the processing toolbox : SAGA → Vector<->raster → Raster values to points) that will convert your DEM to point with the elevation in a field (you need to choose the "nodes" option otherwise the "cells" option give you polygon).
But if your aim is only to see the density of points in your zone of interest you dont need to do anything. Converting a raster to point will give you a point for each cell of your raster so the density of point will be relative to the raster cell size (for exemple if you have a 25m cell size (square cell) you get points at 25m from each other (one point for 625 square meter or 0.0016 point/m²). So you only need to look at the pixel size in the raster property to get the information you seem to look for.
Also be avare that the DEM you get may have been created from a different resolution...
EDIT : it seem I misunderstood your request. but the SAGA tool "Raster values to points" will still help you for first step you could then use the plugin "DataPlotly" to create a scatter plot in qgis or export the attribute table and use any other tool