No, there is no formal mechanism in Minecraft for specifying and tracking player-defined goals. Some things players do to create their own goals include:

Challenges - Survive for X days, collect Y resources, build Z structure, etc.
Achievements - Create a list of tasks to accomplish and check them off as completed
Adventures/Quests - Come up with a story/narrative and goals to progress through it
Builds - Plan and execute large construction projects

But all of these are tracked externally by the players themselves. There is no way to formally specify goals in the game and have them tracked/triggered automatically when completed. You would need to communicate goals to other players verbally or via other external means.
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