As I said in my comment, for non-referenced images is preferable to use The Gimp; not QGIS. However, if I use only The Gimp in my answer, I think there will be no implicit GIS component. Therefore, I will try to refer to QGIS from here on. First of all, I loaded in QGIS a non-referenced image (png format) that represents a Cylindrical Equidistant Projection of world (2048×1024) centered on the Pacific Ocean. It can be observed as follows:
By using The Gimp, I crop above image in four .png images of identical size (1024x512) following clock-wise sense (named img1, img2, img3, img4; respectively). As in your situation, when I load the 4 images as raster layers in QGIS, they all overlap, since they are not georeferences.
However, I know that original image has this reference in WGS84:
xmin ymax xmax ymin
0 90 360 -90
So, by using gdal_translate command in OSGeo console of QGIS as follows:
gdal_translate -a_srs WGS84 -a_ullr 0 90 180 0 img1.png img1.tif
gdal_translate -a_srs WGS84 -a_ullr 180 90 360 0 img2.png img2.tif
gdal_translate -a_srs WGS84 -a_ullr 180 0 360 -90 img3.png img3.tif
gdal_translate -a_srs WGS84 -a_ullr 0 0 180 -90 img4.png img4.tif
I got four referenced .tif images. They can be merged without any problem. This merged image, completely indistinguishable of four individual images, it can be observed in following image.
You can adapt this approach for referencing and merging the 4 images of Mars surface.