Galaxy mergers in clusters explain the large central galaxies in clusters. In fact, the centers of clusters often host a special type of galaxy called a cD which is usually extremely massive because of galactic cannibalism. Dynamical friction makes galaxies spiral down toward the center where they get eaten by the central galaxy and that in turn makes the central galaxy more massive and more efficient at this process.
In the rest of the cluster, the velocity dispersion is significantly higher than the escape velocities from individual galaxies. So a) galaxies cannot gravitationally attract other galaxies because they go by too fast and b) even if there is a head on collision by sheer coincidence, they would pass through and not capture one another. There may well be huge consequences to both galaxies, such as stellar rings and shells formed and tidal tails extracted, but the two remnants would separate and perhaps never meet again.
This is an oversimplification though because clusters often have subgroups of galaxies which have lower velocity dispersion wrt each other and these could merge much as galaxies in groups merge. But the tidal field of the cluster actually works against this and so the rate would be lower.
There is a lot of literature on cD or central galaxies since it is one of the standard types of galaxies. Here is a recent one:
"Growth of brightest cluster galaxies via mergers since z=1", Claire Burke and Chris A. Collins, 2014, MNRAS, v. 434, p. 2856.
A study of merger rates in the outer parts of clusters may take a major effort of going through the results of N-body simulations on your own and your answer may depend on which simulation you use.