Short answer: No, but also maybe, depending on what “destiny” means.
I want to. I’m so filled with anxiety over every decision I have to make, large and small, that it would be a great relief to know my whole life’s been planned ahead of time and I’m only playing a part that’s been written for me.
Also, I’m monumentally lazy, so the idea that I was destined to lay around all day today doing nothing would, again, be a huge relief. I could sleep peacefully tonight knowing I was not, in fact, supposed to do something better with my one precious life.
Finally, I believe the world and everything on it will be incinerated by the sun in ten billion years, give or take a few, so in the sense that the earth and everybody on it has a very predictable end that’s literally written in the stars (well, one star, in this case), that’s sort of a destiny.
But while the idea of fate and/or destiny appeals to the anxiety-ridden/lazy sides of me, I simply can’t believe we have a destiny to fulfill beyond “be nice to other people.” It would be a pretty great destiny and not really that easy for us to fulfill, when you consider the history of humanity, but I doubt that’s what people mean when they ask if I believe in destiny.