I guess that you've got different polygon layers for the colored areas and another polygon layer with the provinces.
What I would do, first is to filter the provinces layer (right click > filter),
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/eI9z9.png)
selecting the one that you want to work with (double click on the field with the provinces names; equal sign; All button; double click on the selected province).
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/DTaKx.png)
After that, combine all "coloured" layers using vectorial > Data Management Tools > Combine vector layers
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/pJ58j.png)
Select all layers and run the algorithm and you'll get a new combined layer. Now, execute the Difference
tool vectorial > Geoprocess > Difference
with the provinces
layer as the imput layer
and the new combined layer as the overlay layer
.
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/8rBrA.png)
You'll get a new layer with the "uncoloured" polygons. After that, you can create a new field for that new layer, using the field calculator
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/xkD1I.png)
with the following settings
![enter image description here](https://cdn.statically.io/img/i.sstatic.net/OgaPr.png)
You'll get the area of each polygon in the units of the layer.
difference
to substract the coloured areas, tada.