Here you have mixed two different and complex question about 1) how to make tree which shows phylogeny and 2) about analysis in R.

So for the first part, as I understood from your question you want to do phylogeny of primates based on any specific gene and not specific gene evolution in primates. If you want to analyze primate phylogeny, one or two specific gene can not tell you the answer. You have to make species tree in stead of gene trees. And this is not an easy task and I don't think if this can be done in researchgate discussion. You have to invest significant amount of time and here is the perfect link to learn the basic and analysis pipeline http://evomics.org/workshops/2019-workshop-on-phylogenomics-cesky-krumlov/

I did something quite similar for my dissertation and for an MBE publication in 2000. There I was comparing the evolutionary histories of 12 mitochondrial protein coding genes (gene trees) to the concatenation of all mitochondrial protein genes (the presumptive species tree) for 63 mammals including primates. I compared the likelihoods of the gene trees to the species trees with Shimodairo-Hasagawa tests. (A comparison of sequences using a refined paired t-test that corrects for various sources of error including multiple testing).

Each gene tree was different from one another and from the "species tree". But were they statistically different? If they were statistically different than I would have recognized statistically different gene histories that differed statistically from the expected species tree. Any statistically different gene tree would have represented distinct evolutionary histories from the species tree (not uncommon). No single gene retrieved the expected species tree but a combination of three genes (ND1ND2COI) retrieved not only the tree topology but the expected branch lengths.

As it turns none of the gene trees were statistically different and so simply differed randomly from the species tree built from all genes. But these genes are all linked on the mtDNA tree. Nuclear genes can have randomly different genes but also may be systematically (statistically significant) different from one another and from the species tree. This indicates that the gene has had a somewhat different evolutionary history than the species as a whole.

Although other methods exist, a sound method to test for the best gene is to infer the maximum likelihood trees for each gene using an optimal model for each protein-gene (stratifying by codon position) and comparing the resulting gene tree to the one inferred from all genes (stratifying by codon position as necessary). Any gene that is significantly different from the concatenated sequence of all genes is not a good gene for retrieving the best scientifically-known tree.

To compare it to the current "best" species tree for primates (instead of the the concatenated tree if they are different) you will need to constrain the topology of that tree by writing your own Newick file if it is not in the literature. You must make sure that the taxonomy labels match the ones from your gene trees so delete any taxa from the Newick file that the scientifically-accepted includes but you data does not. Then run each in the series of 22 gene trees against your new Newick file with SH tests and see what happens.

Keep in mind that all phylogenies are merely estimates of evolutionary history and every method and every data set has it own biases. Molecular sequences have all the same biases (convergence of characters) that morphological data does so whatever tree you use for comparison will be an estimate itself. Use maximum likelihood whenever possible and as much well-modeled data as you can get your hands on and know that single genes will rarely find the same tree (they are just too short with some notable exceptions.

Let me know if you need some help.