Per the rules, language proficiencies can be taught with downtime training but weapon and armor proficiencies cannot.
Per what seems reasonable to me, there are some things in which animals can't be trained. For instance, even though tool proficiencies can also be taught with downtime training, I wouldn't let the player teach an animal how to use artisan's tools, because it's an animal and it just doesn't have the brain for that. There's no explicit rule saying that you have to be a humanoid or have so much Intelligence to gain a tool proficiency, because the rules are written for humanoid adventurers and such a rule would be a waste of page space in 99% of all cases. So when you're in the 1% case, you've got to make a judgment call.
Now, if you, as DM, do think that it's a reasonable proposition to teach an ape how to fence, go ahead and allow it. I'm not going to stop you, and for all his antipathy to the idea,
@Paul Farquhar can't stop you either. But we were asked for advice, and my advice is that there is nothing in the rules which allows this and it sounds implausible and silly.