This new idea/story includes an ancient subterranean race, but they need to survive harsh conditions. Developing a nervous system/brain is important. Are certain proteins necessary to form a complex brain or can they thrive on volcanic conditions to become sentient? (i.e. carbs/protein vs chemicals, in line with deep underwater life found near volcanic vents) Part of story below, as humans accidentally discover technology from another race indigenous to Earth, a secretive and well hidden species, hence an unknown subterranean world. Hope this question makes sense, as I need scientific explanations for the existence of this 50-million year old race with superior intelligence/technology. https://artidan007.wordpress.com/2016/11/26/black-knight-antarctic-dawn/
It's more likely that animals eating the extremophiles will evolve intelligence.
The problem with extremophiles themselves evolving intelligence is that they tend to function best in very specific conditions. Bacteria that live around hot springs, for instance, don't tend to live outside of a narrow environmental band that they've adapted to. Acidophiles, similarly, won't be found outside of highly acidic environments.
However, there are animals that eat extremophiles that are found outside these areas. Some marine worms, for example, eat bacteria that thrive in extremely high temperature environments in and around undersea vents. These worms generally keep the respiration and important bodily functions end of their body in relatively cool water, with only the mouth end in the high temperature environment, allowing them to thermoregulate their bodies to some extent and not die.
These worms are still considered extremophiles, but the animals that eat these worms may not be. In an environment with a sufficiently rich food source based on extremophiles living in a chemically rich environment, it's possible that an ecosystem containing life forms significantly more complicated than worms may arise with extremophiles as the bottom of the food chain, but not the entirety of the food chain. The predator animals, probably predators towards the top of the food chain, would have reason to develop intellect and tool use as a means of attaining food supplies that fight back more than things like sulfur or bacterial mats.
If there were a potent, consistent source of energy deep beneath the Earth supporting an extremophilic population, it's possible that your intelligent creatures could have evolved as part of an ecosystem completely disconnected from the surface. For example, they could live on a uranium deposit which bacteria like desulforudis audaxviator have evolved to utilize as a food source. If these bacteria exist in sufficient numbers, they could serve as a food source for shipworm-like creatures that burrow into the uranium mass, while keeping their reproductive systems out of harms way. Everything else in the ecosystem could ultimately feed on these creatures, with your sentient race somewhere near the top of the food chain.
Without any connection to the surface, and without a need to move away from their food sources into the radiation-free rock above, it's possible that an intelligent race could evolve without any knowledge of the surface. However, it's unlikely that they would develop technology on the level that we possess and remain strictly underground. Intelligent creatures would likely develop technology and scientific theory that would lead them to learn that they live underground on a spherical planet. Such creatures would have no reason no to want to at least explore the surface of the Earth, even if they have no reason to believe that life could live there, away from the uranium beds. By the time they reached our technological level, they'd certainly have drilled up to the surface so see what's up.
Evolution by its nature changes the design life around issues in environment. Considering that life manages to thrive in almost all environments, I'd say the biggest difficulty with extremophile intelligence is less dependent on chemical requirements and more on actual competition in such a harsh environment.
Modern theories regarding the evolution of human intelligence point at harsh conditions and prey status as strong catalysts for advancement. The points of major relevance can be distilled to three reactions to predators by humans that drove them to higher intelligence:
Better situational assessment - Early Homininae reacted to increasing exposure to predators by leaning towards bipedalism which allowed them to see more of the environments due to increased height
Handedness - Those same Homininae soon took to handedness as a means of wielding the environment against their predators
Brain Size increase - As the Homo genus began to emerge, brain size developed with them. This allowed the hands to utilize fine motor control for the construction of tools used to best the physical more capable predators in nearly every condition. Also, the brain allowed development of communication between individuals that allowed very precise coordinated efforts when tools alone would not suffice.
Assuming your environment creates and maintains a predator of sorts that requires the intelligent species to evolve to be more intelligent to handle it, you have a fine method of reasoning. To following up on Samuel's comment, another issue is explaining the secretive existence of the species. Perhaps a system of religion or even an instinct that evolution created to drive the species to where they are and keep there would suffice.