During the p.adj , many measurements (different genes) from the same sample (i.e one column of the image) does it take all the p values from across all the samples and rank them? or ranking gene by gene and calculate the p.adj?

For example, if we have 100 genes that we want to test for differences between three samples A, B and C, then do we generate 100 p-values for A x B, 100 p-values for A x C and 100 p-values for B x C, for a total of 300 p-values? and adjust all 300 p-values at the same time?

Does that mean all the 300 P-values rank together and do the p adjustment for all the samples (A,B,C) treating as one population?

OR, do we adjust the p-value each sample by sample (column by column)

I have used Bioconductor dep (limma dependent) package for data analysis and for one sample I get p.adj = 1 for all the genes (sample output attached).

If you use limma, then all the procedures are explained in the limma documentation. By default, limma adjusts the p-values for each comparison separately but limma offers a number of other options as well. If you do three pairwise, comparisons, then limma can adjust each separately or all at once depending on what you ask for.

If you use DEP, then you need to ask the DEP authors. The output you show is not standard limma output. I don't know what DEP does or what code you have used to call it.

Usually, getting p.adjust=1 for all genes simply means there are no DE genes for that comparison, but the output you show seems wrong. If I am reading the output correctly, then you have a case with p=4.2e-14 and adj.p=1 for the same peptide and the same contrast. That is impossible.