In 2 Kings 18.26-28 [paralleled by Isaiah 36.11-13], 2 Chronicles 32.18 & Nehemiah 13.24, there is mention of Yehūḏîṯ, which literally means "Judaic," "Judaean" or "Jewish [speech]": the language of the kingdom of Yehudah [Judah], which we might say is Hebrew, but which could alternatively be a dialect specific to the tribe of Yehudah (cf. Judges 12.1-7). At any rate there is nothing in any of these passages about God speaking or writing, nor is the term "Hebrew" ever employed (although it is [mis]translated to say that in a few English Bibles).
As a matter of fact, in the Bible itself (at least the parts thereof considered to be canon by most Christian groups) the first time that Hebrew is mentioned as a language is actually in the New Testament.1 There is no direct or explicit reference anywhere in the Old Testament to a language called Hebrew nor is there even a phrase like "the language of the Hebrews" which might perhaps at least imply such a phenomenon.
A Babylonian in the Land of Canaan
Variations of the term "Hebrew" appear 34 times in the Old Testament, always in reference to a person or a group of people. The first such instance is in Genesis 14.13, where Abram is called "the Hebrew", which at the time was not an ethnonym, Abram himself being a Babylonian (or "Chaldean")2 and having friends or acquaintances among the Egyptians (he was a guest of one of the Pharaohs, according to Gen. 12.10-20), the Philistines (he cut a covenant with a Philistine king, Abimelech, in Gen. 21.22-34, to which king God spoke in Ch. 20), the Hittites (from whom he bought his family tomb in Gen. 23) and the Canaanites (consisting of roughly ten goyim/ethnoi, "tribes/nations"3, in whose land he dwelt for the second half of his lifetime).
It might be safe to deduce from the above that Abram/Abraham was multilingual, able to speak Akkadian (Middle Babylonian in his case4) and Hittite; the Proto-Arabic of ancient Canaan; and the languages of the Egyptians (perhaps Middle Egyptian at his time) and the Philistines. It may be that he did not even know of a Hebrew language. The alternative to this would be to assume (and it's just as much an assumption as is any other theory) that Abraham necessarily chose to speak Hebrew (wherever he would have acquired it from) and that all these different peoples he encountered necessarily spoke his language of choice, including his Amorite comrades Mamre, Aner and Eshcol (see again Gen. 14.13), and also his God.
The Slave Who Named God
Abraham's wife Sarah had an Egyptian slave named Hagar, who is the first person in the Bible to ever give God a nickname. In Gen. 16, Hagar also bestows upon the well between Kadesh and Bered the presumably Hebrew name of Be'er-Lahai-Roi, the "Well of the Living One Who Sees Me," because there she had seen God and spoken to him. One might conclude from this that the conversation between God and Hagar took place in Hebrew, but it could just as well have been conducted in Hagar's native Egyptian dialect and, after her naming of the well, either she or its later users translated the name into the local Canaanite dialect (from which eventually Hebrew might have originated).
Diverse Cultural and Ethnic Origins
In Gen. 42, Joseph, a great-grandson of Abraham and Sarah, meets his brothers in Egypt years after they have sold him into slavery. V. 23 of this chapter indicates that Joseph's brothers, who think he is merely an Egyptian, are also under the impression that Egyptians typically do not understand their speech. Joseph helps this assumption along by necessarily using a translator to communicate with them. There is, however, no mention whatsoever of what language is being spoken by anyone in the scene, nor, for that matter, by anyone in Egypt or Canaan.
A few generations after Joseph, in the Book of Exodus, it is in this East African environment, filled with points of contact with foreigners of various stripes, that Moses and his fellow Israelites find themselves. If Moses was indeed raised by and as Egyptian royalty5 it stands to reason that he spoke and wrote Egyptian quite fluently. (The Ancient Egyptian name for their hieroglyphical script is Medw Neter, the "Words of God."6) If he didn't already speak the language of the Kenites of Midian before he married the daughter of their priest7 then surely after forty years living among them8 he must have become quite well-versed in that tongue as well. And if Moses' Kushite ["Ethiopian"] wife mentioned in Numbers 12.1 is the same person as the Kenite's daughter, then the Kenites themselves seem to have been a diverse and—it would stand to reason—multilingual population, possibly best described as Afro-Arabian (not unlike many Northeast African, South Arabian and South Indian ethnic groups of the modern era).
Leviticus 24 contains the story of one of the Israelites who is sojourning in the wilderness between Egypt and Canaan during the Exodus, whose father is Egyptian and whose mother is from the tribe of Dan. Lest that be taken to be quite the peculiar anomaly, note too 1 Chronicles 4.18, in which the daughter of a Pharaoh is married to an early descendant of Judah! The patriarch Judah himself was married to a Canaanite (Gen. 38.2) and Joseph's wife was the daughter of an Egyptian priest (Gen. 41.45).
So Israel was itself a diverse population both at its onset and at the time of the Exodus. Moses himself grew up as an Egyptian, married into a family to which he was only very distantly related through Abraham and which lived a good distance outside Moses's native Egypt (perhaps outside Africa altogether) and he encountered various peoples on his journey with Israel through the wilderness.
There is no part of the Torah/Pentateuch or any other portion of the
Old Testament that indicates God inscribing anything anywhere in
Hebrew (nor in Arabic or Medw Neter or cuneiform or any specific
language). That conclusion is an assumption, which may be quite
correct but is not based on anything that the Bible actually says.
Neither is there any mention in the Old Testament of what language God
ever speaks to anyone when he does speak, neither on earth nor above
Why Necessarily Only One Language?
Going by the demographics of the people who Moses led out of Egypt, let alone Moses' own cultural background and experiences, it should be fair to presume that the stone tablets with the Ten Words themselves catered to a few different dialects or scripts, somewhat like the Rosetta Stone (written in Medw Neter, Demotic Egyptian, and Greek) or the royal Afro-Arabian inscription known as the Monumentum Adulitanum (written in Ge'ez, Sabaic and Greek).
Considering the fact that, according to the story, it is the Deity himself who authored the stone tablets, I don't see why they couldn't have contained the Ten Words copied into ten different languages, or even seventy-odd dialects, symbolic of the traditional number of goyim, "nations/peoples," descended from Noah in Gen. 10.
And when God introduces himself to the shepherd Moses on Mt Horeb, he might be speaking to him in Kenite or Egyptian just as much as Hebrew, or a combination of the above, or perhaps Moses is experiencing a form of the New Testament's glossolalia, hearing the Deity's voice in a tongue completely unknown to him but which he is nonetheless able to comprehend and to converse in.
Bala'am of Mesopotamia
In Numbers 22 God speaks to the non-Israelite soothsayer Bala'am, who may even be from the same region of Asia as Abraham was. There shouldn't be much reason to expect that this conversation was definitely in Hebrew, but then again perhaps Bala'am was a Hebrew-speaker and God chose to communicate with him using that tongue.
Only One Tenuous Instance (in the NT)
In the "canonical" Bible the closest to a clear and explicit mention of anything close to God directly speaking to anyone in a particular language occurs in Acts 26.14, wherein the apostle Paul/Saul is making his case to King Herod Agrippa. Herein he tells the king that when Jesus accosted him on his way to Damascus, he questioned him "in the Hebrew dialect". Curiously, however, a few English Bible translations such as the NIV, the NLT and the NET disagree with the Greek texts by saying that it was "in the Aramaic language". Surprisingly, even the TLV, a Messianic Jewish Bible, says "Aramaic". The "Aramaic Bible in Plain English" translation says "Judean Aramaic". The ancient Syriac Aramaic Bible, called the Pəšîṭtâ, says "Hebrew" just like the Greek New Testament and most English translations.
1. See e.g. Luke 23.38, & John 5.2, 19.13-20, & 20.16
2. See Genesis 11.31
3. See Genesis 15.18-21
4. Although Alice C. Linsley argues that it was "Kushitic" [East African] Akkadian.
5. Exodus 2.10
6. See also Acts 7.22
7. Exodus 2.16-3.1, & Judges 1.16 & 4.11
8. Based on Stephen's opinion in Acts 7.23 read together with Exodus 7.7.
9. Am I allowed to quote myself? Heh-heh... :-D