Besides quidditch, The Wizards and Witches played some other games too, like Chess and Gobstones (that's what i think is the spelling, Saw it in HP5 game). So is there any fact of World Cups or even Inter House Tournaments being organised of other games/sports?
Gobstones has organised teams and tournaments, both professionally and at Hogwarts.
We first learned about this from the first issue of the Daily Prophet that JK Rowling had written for Bloomsbury's official Harry Potter fanclub, which included a letter that referenced the Welsh National Gobstones Team and they're victory over Hungary.
GOBSTONES TOURNAMENT OVERLOOKED
As a member of the Welsh National Gobstones Team (...) I was surprised to see that the Daily Prophet did not cover our resounding triumph (...) I was most upset to learn that our victory was not deemed worthy of a single line in your paper.
-- Grugwyn Rufford
OUR EDITOR WRITES
I am sorry to hear that you do not feel we give enough coverage to the game of Gobstones. The reason for this is that most wizards find it deeply boring.
In book six we learn that Hogwarts had a Gobstones Team:
The picture showed a skinny girl of around fifteen. She was not pretty; she looked simultaneously cross and sullen, with heavy brows and a long, pallid face. Underneath the photograph was the caption: Eileen Prince, Captain of the Hogwarts Gobstones Team.
The Pottermore article on Gobstones goes on to mention The National Gobstone Association.
Professional Gobstone players compete in national leagues and international tournaments, but it remains a minority sport within the wizarding world, and does not enjoy a very 'cool' reputation, something its devotees tend to resent. Gobstones is most popular among very young wizards and witches, but they generally 'grow out' of the game, becoming more interested in Quidditch as they grow older. The National Gobstone Association has attempted recruitment campaigns such as 'Give Gobstones A Second Glance', although the choice of accompanying picture (current Gobstones World Champion Kevin Hopwood being squirted with an eyeful of gunk) was perhaps ill-chosen.
Yes, there are professional competitions for other sports.
But Quidditch is by far the most popular and well-respected. Some examples:
Quoting from the Pottermore article about Gobstones:
Professional Gobstone players compete in national leagues and international tournaments, but it remains a minority sport within the wizarding world, and does not enjoy a very 'cool' reputation, something its devotees tend to resent.
Possibly a tongue-in-cheek reference to the state of real-world US sports (which often differ from their international counterparts), Quidditch Through the Ages mentions a rival sport named “Quodpot”:
The United States has not produced as many world-class Quidditch teams as other nations because the game has had to compete with the American broom game Quodpot. […] Quodpot has had some success as a minority sport in Europe, though the vast majority of wizards remain faithful to Quidditch.
It seems likely that there are national tournaments for Quodpot, and that there are Quodpot competitions at the American school of magic (the name and structure of which is as yet unknown).
Moving quite far the canon stack, Harry also meets Chess champions from the different houses in the Order of the Phoenix game – there must be some sort of inter-house tournament. Although given Ron’s skill in the game, it’s a little surprising that he never seems to be invited to take part.
There are many sports and games in the wizarding world besides Quidditch, however, it is the most popular.
I assume that this is what you are looking for, as you mentioned Quidditch. I've just added the former as you included "Games" in your title.
There are other games, clubs, and organization for games but Quidditch is the most popular. We learn one of the possible candidates for Beater on Angelina's team said that if it interfered with his/her's (can't remember) gobstones team they wouldn't play. We also learn that Snape's mom was the head of some sort of team in the newspaper. I don't think there is a direct canon, but I find it most likely there are official and international World Cups for these groups.