My landlord’s dog is very excitable medium-sized terrier mix of some sort.
I like to give him treats every now and then, but the guy gets SO hyped as soon as he sees me going for the pack of biscuits that he starts breathing weird, and a couple of times it sounded like he was choking and began coughing.
How can I reduce his enthusiasm? (for his own good, I feel bad denying him treats because I’m worried about him choking).
The other side of this is that he’s a somewhat nervous fellow too, I don’t want to scare him in any way.
The classic dog-training technique is to simply wait until he’s calm, then give him the treat. If you feed him while he’s acting up, you’re just reinforcing that behavior. If he’s trained at all, you can tell him to sit or something - anything to get him calm and quiet before you reward him.
That said, some dogs are just really highly strung. We’ve worked a lot with our Jack Russell, and he’ll generally listen and do what he’s told, but there are times when he’s just in his own little world and not even aware of our existence.
Without seeing how the two of you interact during the dog’s episode, this is a difficult question. As an example, if when you first meet, you say something to excite the dog like, “Do you want a treat!”, then naturally the dog’s excitement will increase. If you say nothing, then the dog may simply become excited in anticipation of the treat which is natural.
Retraining a dog takes both perseverance and patience as well as trying several tactics. Since, when the dog first sees you it begins thinking “Treat Dispensary Human Has Arrived!”, it gets excited. I have no idea what the coughing/chocking symptoms may be and suggest you ask the owner. It may be reverse sneezing which mimics a chocking sound and some breeds do this when excited.
My approach would be to stop giving treats for the time being. When you see the dog (try this with the owner present), calmly approach it without speaking or making eye contact. Once you’re next to the dog, stand there quietly without speaking or looking at it until it calms down on its own. This may tale 30 seconds or it could take several minutes but eventually the dog should calm down once it realizes you’re not handing out treats.
Treats are a way to motivate dogs but currently when the dog sees you it is motivated toward excitement and you reward and encourage that excitement by giving it food and this increases the motivation to continue and perhaps even escalate that behavior.
When it does calm down, at that point you ‘reward’ the dog by looking at it and petting. If it gets excited again, ignore the dog until it calms down once again. This approach may have to be repeated several times until the dog understands that your arrival means it needs to be in a calm state of mind before it gets your attention. Dogs are pack animals and they crave attention.
In addition to the above - you could carry a dog biscuit in your pocket, palm it and give it to the beast before it has time to get worked up into a mini-frenzy.
He didn’t get a treat, and threw a tantrum instead
Excitable dog, they all said
And he bit the tenant’s leg when he didn’t get fed
Excitable dog, they all said
Well, he’s just an excitable dog
Kick it in the freckle.
In addition to the good advice upthread you can try to desensitise him to the excitement of treats. Every so often approach the area where the treats are but don’t give him one. Build it up so that you handle treats or things in the treat area but he gets no reward. Eventually he’ll learn that you don’t necessarily give treats every time you pick one up so there’s no need to get all excited cos who knows if he’s going to get one or not. Extend this by teaching him that calm behaviour earns a treat.
Go get a treat, let him get excited but then don’t give it to him. At some point he’ll calm down a bit (that may take a while) at which point you can give him the treat. Or not, as you choose. Retaining the treat now and again can reinforce your higher status as Handler of Food. Some dogs get very excited every time they see their humans put coats on. They’ve learned that coat on leads to humans going outside which often means an exciting walk. The trick there is to put on coats, wait a short while and then take them off again. The stimulus of putting a coat on eventually becomes meaningless. The training can be extended to putting a coat on and going outside only to come back in a minute later as though nothing had happened.
One thing it took me a long time to realize, is dogs are happy with a pat on the head. They love food but they are just as happy with a cuddle and a pat on the head.
Anyone remember the Saturday Night Live sketch from the 70s, that featured the commercial for the products “Puppy Uppers” and “Doggie Downers” 
For an older dog, it might not work, but what I do is make a “calm down” sound and wait for the dog to obey (like what was suggested above, but I use an aural prompt as well.) My dog now is trained to calm down when I clap my hands.