The three legendary birds Articuno, Zapdos, and Moltres can also be found in Pokemon X and Y. Which legendary bird you can catch is based on which Pokemon you chose as your starter. Articuno will appear if you chose Chespin, Zapdos if you chose Fennekin, and Moltres if you chose Froakie.
Zapdos can be found roaming the wild; after the first encounter, the Pokedex can be used to locate it. Zapdos has been spotted in the following Routes after beating the Elite Four: Route 22, Route 18 (can encounter even while using repel).
Articuno can be found roaming in the wild, after the first encounter the Pokedex map can be used to locate it. Articuno has been spotted in the following Routes after beating the Elite Four: Route 19, Route 15.
Moltres can be found roaming the wild, after the first encounter the Pokedex map in the Coastal Region can be used to locate it. Moltres has been spotted in the following Routes after beating the Elite Four: Route 18, Route 4, Route 5.
Once you Encounter them once, they will instantly flee, not giving you a chance to fight them. You must track them down, like you did with Entei, Raikou, and Suicune. However, they will continue to flee for a total of nine times. After that, they will be at the Sea Spirit's Den, where you can battle and capture them. They will be level 70.
An easy way to enounter them enough times for them to appear at Sea Spirit's Den is to stock up on max repels and have a Pokemon under level 65. Fly to Santalune City and spray the max repel. Next head over to Route 4 and run in the grass. You will either find no Pokemon or you will run into one of the legendary birds. If you do not run into one of them go back into Santalune City and then immediately back into Route 4. This will cause the Pokemon to switch between the route it was previously and go into a new random route. Continue this process until you run into the Pokemon enough times for it to make it's way to Sea Spirt's Den where you can capture it. Remember to have a Pokemon under level 65 at the front of your party because if you have any higher the repel will repel the legendary Pokemon away to.
Also note that if you use Fly or one of the trains the location of the Legendary Bird in your game will change.
Hope I helped :)