Splitting polygon using another polygon in QGIS?

It's simple

Step One - "Vector> Geoprocessing> Difference";

Step Two - “Analysis Tools> Split Composite Objects” (This step is necessary in order to convert a multipolygon to a polygon, otherwise the result may not be accurate);

the third step "select a layer in the layers panel with the name" Single_parts ", click on the right mouse button and in the pop-up window set the checkbox - Show number of objects" and this is your result ...

The easy way

There is a very easy way to do so:

You can use Vector -> Geoprocessing Tools -> Clip.

Input Layer is the underlying, greenish one.

Overlay Layer is the clipping, orange one.

This will create a new layer (and if you leave the Clipped name empty, it will default to a temporary, disposable layer, so you don't pollute your hard drive).

The number of individual features of the clipped layer is the number you are looking for.

The flexible way

You can avoid doing the manual count (useful if you have a large number of features) by using:

Toolbox (the "gear" icon) -> Join attributes by location (summary)

and inputting something like in the following screenshot:

Join attributes by location (summary)

Please note that, depending on the options you select, you can yield many more useful insights about the newly clipped polygons ;)