In the real Middle Ages
"In my world, the main empire I'm focusing on (of late Classical to Early Dark Ages technology level) has begun expanding on a continent the size of Eurasia. Explorers cross a large mountain range, and discover a large area of forest."
Great! We have a nice historical example. The time is the 11th century. The "great empire" is the Kingdom of Hungary, the "large mountain range" is the Southern and Eastern Carpatians, and the "large area of forest" is, well, most of territory which would eventually become Wallachia and Moldavia.
The only difference is that there were no "explorers", because they were not needed. The geography of the land was very well known, and had been known for a very long time.
So what happened in the real history?
In the 11th century, nothing. The kingdom of Hungary sat on their side of the Carpathians.
In the 12th century, nothing. The kingdom of Hungary sat on their side of the Carpathians.
In the 13th century, nothing. The kingdom of Hungary sat on their side of the Carpathians.
In the 14th century they attempted to establish suzerainty over Wallachia; mind you, not to conquer, just to establish suzerainty. By that time it was way too late; Wallachia had become an organized state, and Basarab I of Wallachia defeated the expedition of Charles Robert of Hungary.
Why was it like this? Why didn't Hungary expand into the power vacuum to the south and east? And this is not the only example in European history; remember, for example, that the entire (densely forested) region south and east of the Baltic was only conquered (by the Teutonic Knights) in the 13th century; before that, the Baltic people were left alone to fend for themselves.
There are two fundamental problems with envisioning western European medieval "empires" (note the scare quotes) hell-bent on territorial expansion:
First, the perennial problem of western European medieval states was lack of people. There was a severe post-classical population crash in Europe; it took Europe one thousand years to recover the population level it had in the 2nd century, which was not very high to begin with. Land they had aplenty. Forest they had aplenty, and they were actually offering incentives to people to clear it. There was forested and uncultivated land in England, in France, even in Germany, not to mention the large truly wild areas of Scandinavia, of Poland, of the Baltics, of Ukraine, and of western Russia. Western Europe had more than enough land for its people at home, they did not need more land.
Second, there was no way to trade extensively overland; they simply did not have the transportation technology. A mountain range was a very effective barrier to trade. Oh, they could trade luxury goods, small enough and expensive enough; but bulk goods? Timber, grain, wine? Not overland, they couldn't.
Remember the exalting and sad story of the Crusader States: established in a glorious war of conquest in the late 11th and early 12th century, and lost in the late 13th century. The Kingdom of Jerusalem, the Principality of Antioch? They never managed to bring in enough colonists to become sustainable; they never reached a population level sufficient to defend themselves against the Saracens. And why was that? Because Europe itself did not have enough people; sending masses of colonists overseas was out of the question.
In conclusion, it is not at all surprising for a western European kingdom to just sit on its side of a moutain range, beyond which is a dense forest inhabited by dirt poor people. It is to be expected.
If you want the "great empire" to seek to conquer the forest land you must give them a strong incentive; I suggest that instead of "hunter-gatherers" you make it inhabited by a splendiferous civilization, swimming in gold and silver. Then the empire would be interested. But poor peasants they already had; land they already had; forest they already had: they had no reason to embark in costly attempts at conquest and colonization.
The question states that this "medieval" "great empire" is actually in another world. The issue is then, what makes this "great empire" medieval?
The western European medieval society was a historical phenomenon very highly specific to western Europe. It is extremely unlikely to get anything quite like it in another world.
Think about it.
The western European medieval society is a historical aberration. A highly advanced, highly literate, fully functional civilization collapses due to basically a long streak of historical bad luck. To bring about the western European Middle Ages, you need: the Crisis of the Third Century, during which a previously functional empire finds itself mired in a century of civil war; plus a few centuries of barbarian invasions; plus a devastating plague; plus the emergence of Islam bringing the collapse of the (Eastern) Roman Empire.
There were feudal societies elsewhere, in Persia, in the Ottoman Empire, in India, in China, in Japan, and those lands had their own medieval periods; but the western European medieval society, with its highly elaborated double hierarchical system, secular and spiritual, was one of a kind.