As @Babel has suggested, Raster calculator is the best tool.
For your case, there is also a useful (and relatively new) Raster boolean AND tool in the Processing Toolbox > Raster analysis group.
This is a small example, E100N90 is a background SRTM topo (sea level 0 m is set to nodata) and Rectangle is a hand-drawn reddish rectangle. You will notice Raster boolean AND tool at the right-hand side is highlighted yellow.
Just run Raster boolean AND tool and select these two layers as input. Use one of them as the reference layer. The Output layer shows the intersecting area as 1, while non-intersecting area is given 0.
You can then use this layer (commonly called Mask) to extract the intersecting area from two input layers by the Raster Calculator.
Multiplying the DEM (E100N90@1) with the mask (Output layer@1) (see above expression) will return the DEM for the "intersection".
(Likewise; "Rectangle@1" * "Output layer@1").