Merlyn covers it pretty well. They can be synonyms, depending on the context. I just wanted to add that "grief" used in this way is more of a slang, informal way of using the word. Usually the word describes *serious* emotional pain. Like, the kind of pain you feel when a family member dies. So it's kind of a weird word. Sometimes it can mean the worst kind of problem you will ever experience. And sometimes it can be used to describe tiny annoyances. It's become popular in internet slang to be used a verb. When an internet troll (someone who is rude and mean online) is bothering somebody else for no good reason, they are "griefing" that person. Normally you would spell it with a v (grieving). Griefing is only used to describe this kind of internet slang rudeness.

If it is in the context I think you mean, they are very similar but "grief" can mean more like "complain", "He gave me a lot of grief about the grammar mistakes in my report." or "pain" "My back is giving me grief again today." trouble has the same meaning but can also mean more of a physical action (grief can be used this way but it is less likely, at least where I live) "Some bullies were giving me trouble on my way to school yesterday."

Trouble (n) is a very general word covering problems misfortune, worry or inconvenience. To be "in trouble" means you have done something and are now facing the consequences. "He is in trouble with the headmaster for misbehaving in class." "There are many troubles weighing on your sister's mind." To trouble (v) means to worry or inconvenience someone. "May I take your coat?" -----"Don't trouble yourself. "Could I trouble you for a glass of water?" Grief (n) is the emotional response to losing something or someone. To grieve (v) is the process of feeling this emotion. "He was lost in grief after the death of his wife." However, there is a cross-over --- "to come to grief" means to get into a situation that will cause regret, unhappiness or harm. To give _____ grief" means something is behaving in a way to cause worry, anger unhappiness or physical pain. "The boss was giving me grief because I did not file my reports." (the boss is angry and perhaps telling you off). "My arthritis is giving me grief." Hope that helps :) It's all I can think of right now.