According to this article, it would take tens of thousands of years to travel to the nearest star given today's technology.

My question is, what could give the countries of today, or most of them at least, a strong enough reason to work together and build an extremely expensive spaceship and send a group of people (less than 500) to the nearest solar system?

I don't want to worry about how this colony will survive for thousands of generations or what technologies would be implemented to do so.

Because we're human & it's there

We started off in the middle of the desert, yet we kept looking at the horizon and wondering what's there so some of us risked their life & limbs (which at that point really was all they had) & walked there to find out.

At some point we reached the ocean, and we got stuck for a while, that is until some clever cave man figured out that a tree that fall to the water in a recent storm floated, he stood at it and it still floated, suddenly the ocean, a mass body of water that no man can live in longer then he can swim (IE not long) and where every failure was deadly was possible, he didn't know what was behind the ocean (or even if there is something beyond the ocean) but that didn't stopped some of them to take the risk and get on fallen trees and try they luck.

Time went by and the trees became large ships, crossing the ocean wasn't a big deal anymore so we started to look below it, sticking ourselves in what is essentially metal coffins just to go and look what the bottom of it looks like.

But even that wasn't enough for us, birds can fly, we wondered what that feels like so we created planes to let us fly alongside them, it may have other uses now but at first it was just to fly for flying sake.

Then we reached space, the final frontier, and like every frontier before it we are now there as well... not far away yet but our wanderlust knows no bounds, we have footsteps on the Moon & robots on Mars, we are actively working (and spending billions) all around the world (and working together, just look at the international space station) to colonize both in the not so distant future (sure most plans are still in the design stage but even the wright brothers started out that way).

I promise you that once we get to a point we can have a human being live full time on another part of the solar system (which given your question states there's no need to worry about the tech needed I assume it exists in your world), you will still have people looking to other stars and saying "I want to go there", I know that because I am one of those people.

Reason? we don't need a reason, it's there and we're not and that's all the reason we need.

The survival of our species

What happens if we suddenly have a nuclear war and all of us die? Or if the sun explodes? Our species would become extinct. That is why we need to colonize at least a second planet in another star. By that way, we will exponentially increase the survival chances of our species from mass extinction.

War. Survival from other lifeforms

We aren't alone in the universe. It's very likely that there is something there, watching us or taking the galaxy. Just because we can't see any other lifeform it doesn't mean that the Fermi Paradox is true. Maybe all of them are hiding. If there exist several lifeforms in the whole universe, it's obvious that at some point we will engage a war with them for the resources of the galaxy and possibly the whole universe.
We should no longer rest, instead, we must start building massive spaceships to colonize as quick as possible the whole galaxy before others do that! Prepare the ships!

In the near future, probably nothing. In medium term future, there is a number of possibilities.

Today, building a starship like that is simply impossible. But in the near future, when we'd be able to sort out all technical hurdles of sending humans to Mars, a giant interstellar spaceship will become a theoretical possiblity. However, there would still be a number of issues that would make a success of such project highly unlikely.

1. Propulsion. Existing (or practically achievable) methods are too
   slow for interstellar trip, likely making it to last tens of
   thousand years. Even if everything else is done perfectly, we just
   can't expect our generational ship to function that long without a
   failure;


2. Energy. Solar panels won't be effective in interstellar space, and nuclear fission reactors are the only currently available option. This ship would have energy needs much much higher than a Voyager-type probe, which might not be achievable with current tech level;


3. Autonomy. 10,000+ years level of autonomy is impossible to get at our present tech level. 1,000 years maybe is realistic, but still something that we can only hope for;


4. Destination research. Right now we have a very limited ability to detect Earth-like planets in a target systems, and no way of telling if such a planet is a good candidate for colonization;


5. Money. As you have mentioned, this would be a huge undertaking, much bigger than the Apollo project, for example.

So, I would think that even under the best conditions, building such ship in the next 50+ years is impossible. Of course we can try, but the project would have a very low chance of success. I can not think of any realistic calamity that would affect entire Solar system and give us enough notice to build this spaceship.

However, looking further forward, all of the issues in the list above can be eliminated or somehow mitigated. New types of engines can be built, thermonuclear energy become reality, deep space experience would turn into confidence in spaceship's long term durability, our telescopes will find very Earthlike planets and new technologies like space elevator would make building spaceships in orbit very cheap.

Any unavoidable world-ending event, such as a massive meteor hurtling towards earth with a 0% chance of survival. In such an event, the 500 people would be the highest tier professionals in the world in their respective fields, and most likely be people who also designed and built this massive ship, in case of repairs needed. In my head, it has to be something catastrophic to human life on Earth. Nothing less would cause all countries to put money towards any single thing.