Does someone have any experience to produce NDVI on the Landsat1 satellite image? What kind of atmospherical correction (on Landsat 1) technique is necessary to perform before producing indices? Is it possible to compare the values of NDVI between Landsat 1 and Landsat 8 OLI image?
Practically yes! Because both Landsat1 and Landsat8 have near-infrared and red bands. Therefore you might apply atmospheric correction such as Dark Object Subtraction (DOS), a family of image-based atmospheric corrections via QGIS Semi-Automatic Classification Plugin. After atmospheric correction, yes it would be possible to compare, that makes more sense in terms of interpretation. You might apply this technique for both images (Landsat-1 and Landsat-8).
For Landsat-8 images, I would also suggest you to download directly the Level-2 or Level-3 data from USGS (you might demand and afterwards download the data). Both data level are atmospherically corrected. Through that way, you do not need to spend your time to correct image atmospherically. Unfortunately, this service is not available for Landsat-1.