At this point it is safe to say they hate each other, Garp for obvious reasons yet Akainu has this older marine hero who actually he never really respected: we see Aokiji hugely respected Garp as his master yet there are no interactions between Akainu or Kizaru and Garp, yet since Akainu and Aokiji justice are opposites we can guess that just like how Aokiji couldn't bear Akainu and his justice leading the marines the opposite was true as well, so they fought. Then we have Akainu "absolute justice" agenda and Garp who is the father of the most wanted man in the world, the grandpa of pirate jesus plus he also wavered to save the son of the pirate king he adopted... For sure Akainu didn't like that. Also we never had a single word, even just a thought, by Akainu being sorry for killing Garp adopted son (and actually this is almost laughable, it would ruin a bit Akainu best moment because it already clashes a lot with his character).