Many Jamaicans are very generous and warm. Returning this warmth and kindness is a good way to show them that you appreciate their country.
It is likely that you will be asked for money at some point during your trip to Jamaica. Do not feel pressured to give money. The best advice in such situations is to say “I’m fine” and walk away. This also applies to the famous straw markets. The European method of walking away does not work well. You usually have to get involved with someone to get away from them.
That said, if you want to befriend or meet one of the many wonderful Jamaicans and give them a friendly gift, that is perfectly acceptable and welcome. Just use your common sense when it comes to money.
Cultural respect is much more important. You are guests on their island. Also note that you should address older people with “Yes, ma’am” or “Yes, sir”.