Hello & welcome, I am Dr. B, a licensed veterinarian and I would like to help you today.
Now most of our common Bird of Paradise plants will cause GI upset (vomiting, diarrhea, drooling, etc) and lethargy. Usually the fruit/seeds are the most toxic part but we could see GI upset with the stem as well.
Now if he just had this we do have 2 options. If he just ate this and if you think he had quite a bit, then we could induce vomiting now. To do so at home, you can administer 3% hydrogen peroxide orally at a dose of 1ml per pound. (2 teaspoons per 10 pounds of body weight). You can give it via dropper, syringe, turkey baster – we just want to get it in. After giving this orally, move the abdomen around or get your wee one walking about to get things mixing. This should usually lead to vomiting. If it is unsuccessful after 10 minutes then it can be repeated twice more. And if we still have no vomiting, then you'd need to consider seeing your local vet (or ER vet) so that apomorhpine (a very strong injectable emetic) can be administered just get this out of the stomach and avoid any adverse issues.
Otherwise, we can monitor and treat symptomatically for any GI upset here for the next 24-48 hours. To do so, you can consider offering a light diet option for a few days. Examples of an easily digestible diet include cooked white rice with boiled chicken, boiled white fish, scrambled egg, or meat baby food (as long as its free from garlic or onion powder). Ideally, we want to offer this as small frequent meals to keep the stomach settled.
Further to this, if we see any hints of nausea, then we can also treat with an OTC antacid like Pepcid (More Info/Dose @ http://www.petplace.com/article/drug-library/library/over-the-counter/famotidine-pepcid) or Zantac (More Info/Dose @ http://www.petplace.com/article/drug-library/library/over-the-counter/ranitidine-hcl-zantac). These are usually given 20 minutes before offering food to allow absorption and of course you want to double check with your vet before use if your wee one has any know health issues or is on any medications you haven't mentioned. And of course, if he was very nauseous, then we’d want his vet to use injectable anti-vomiting medication to soothe his stomach after this misadventure with Bird of Paradise.