There is a very lengthy article entitled "Jehovah" in the Jehovah's Witness "Insight on the Scriptures", Volume II book, published in 1988, running to 16 pages. Fortunately, there is a heading, "Known by the first human pair" and this is where Genesis 2:4 is first detailed. It says:
The name Jehovah was not first revealed to Moses, for it was certainly known by the first man. The name initially appears in the divine Record at Genesis 2:4 after the account of God's creative works, and there it identifies the Creator of the heavens and earth as "Jehovah God". It is reasonable to believe that Jehovah God informed Adam of this account of creation. The Genesis account does not mention his doing so... Much communication undoubtedly took place between Jehovah and his earthly son that is not included in the brief account of Genesis.
Eve is the first human specifically reported to have used the divine name. (Ge 4:1) Page 13
However, I could find nothing about why the divine name was never used in writing in either the Hebrew or the Greek languages in the Bible prior to Genesis 2:4. That is hardly surprising, though, given that the Jehovah's Witnesses are in agreement with all other translations that likewise don't use the divine name until then. There is no difference between the New World Translation and all other translations with respect to not translating 'Jehovah' or its equivalent (e.g. YHWH, Yahweh) until ch. 2 vs. 4.
Another English translation (nearly 100 years older than the New World Translation of the Hebrew Scriptures) likewise reads:
"These are [births] of the heavens and of the earth in their being
prepared, in the day of Jehovah God's making earth and heaven."
Young's Literal Translation
All Hebrew manuscripts of Genesis only start having YHWH in the text from 2:4 onwards. So, all translations have exactly the same reasons for not using 'Jehovah' until then, for that would be an unwarranted insertion if they used it from 1:1 till 2:3. There is no issue here, no clash between translations.