Likely you have in mind something more difficult, but if it is that simple (checkerboard with two colours, replace with different two colours), you may:
1. click on "select by color tool" and click on one of the two original colours
(you end up will all squares of that colour selected, usually it's recommended to do "select grow" by 1 pixel)
2. having set your replacing first colour as BG for instance, "fill" the selection with BG
3. repeat steps 1 and 2 for the second colour
But maybe also you like to have just one single step...
1. click on "select by color tool" and click on one of the two original colours
(you end up will all squares of that colour selected, usually it's recommended to do "select grow" by 1 pixel)
2. having set your replacing first colour as BG for instance, "fill" the selection with BG
3. repeat steps 1 and 2 for the second colour
But maybe also you like to have just one single step...