Learn Turkish: Bunun konuyla hiç alakası yok. - This has no relevance to the topic at all.

To say "X has Y" or "X does not have Y" in Turkish, we use the structure X'in Y'si var/yok (literally "X's Y exists/doesn't exist"). So, to say "This has no relevance", we must say bunun alakası yok (literally "this's relevance doesn't exist").

In English we say "relevance to the topic". Why does Turkish use the instrumental case (konuyla) instead of the dative (konuya)?

In Turkish, words expressing connection, relevance, or interest—like alaka or ilgi—almost always pair with the instrumental ending -(y)la/-(y)le (with). You are literally saying it has a connection with the topic.

Could I use ilgisi instead of alakası here?

Yes, absolutely. Bunun konuyla hiç ilgisi yok is perfectly natural and means exactly the same thing. Alaka is an Arabic loanword and ilgi is its Turkic equivalent; both are very common and used interchangeably in this context.

How exactly does hiç change the meaning of the sentence?

While yok on its own means "there is not," adding hiç before a negative word acts as an intensifier. It changes "This has no relevance" to "This has absolutely no relevance" or "no relevance at all."