Learn Turkish: Büyükelçilik, ittifakın güvenli olduğuna ilişkin resmî bir açıklama yaptı. - The embassy made an official statement that the alliance is safe.

English uses the conjunction "that" for statements of fact, but Turkish uses a modifying noun clause. Here, olmak takes the factive suffix -dIk and the 3rd person possessive -u to make olduğu (its being). We add the dative suffix -na because the postposition ilişkin (regarding/relating to) requires it. Literally, it translates to "a statement relating to its being safe."

Could I use dair or hakkında instead of ilişkin in this sentence?

Yes, but you have to watch the case endings. The postposition dair also takes the dative case, so olduğuna dair works perfectly here. However, hakkında does not take the dative; it follows the possessive ending directly. So you would drop the dative -na and say olduğu hakkında. Because this is an official embassy statement, ilişkin or dair sounds suitably formal.

Why does ittifak take the genitive ending -ın?

When you create a factive clause using -dIk (like olduğu), the subject of that embedded clause must take the genitive case. Since the alliance is the thing that is safe, it becomes ittifakın (the alliance's). The literal structure is "the alliance's being safe."

Why is there a comma after ?