I have been trying to find an answer to my question for a couple hours now but I haven't had any luck on Google though I might be searching using the wrong terms I'm not sure as I know next to nothing about mining. Anyway, could it be possible for a mine rich in either iron or copper to also produce some gemstones? I don't mean the big ones like diamonds, sapphires, emeralds or rubies but stones like peridot or turquoise or topaz.
Gems such as diamonds, corundums (ruby and sapphire) come from igneous sources. It's possible that one could have a diamondiferous kimberlite blast through a gold or silver ore-bearing formation, or an intrusive dyke that produces rubies or sapphires to penetrate into a similar formation. It's unlikely, given the relative rarity of the two different commodities, but certainly possible.
As an example that was a near-miss (on a continental scale), the Jericho kimberlite in Nunavut (mined for a few years) is only about 30 kilometers from the Lupin gold mine, and there have been kimberlites in that same area found even closer to the mine.
Amethyst, as another example, is just coloured quartz, and quartz veins provide the source of gold in some mines. I'm not aware of amethyst specifically being found in any gold mines, but it's at least theoretically possible.
For 17 years I was involved in writing agreements concerning minerals rights and royalties and those agreements had provisions to cover the possibility of gemstones being found and produced at a mine for a different commodity.
As you mentioned, there doesn't appear to be a lot of information about this online, but it does seem some gemstones are associated with different metal ores. Quartz is associated with gold and a few other types of veins, and forms a variety of gemstones (amethyst, citrine, rose quartz, etc.), but I don't know if quartz veins actually contain any useful gemstone-quality pieces. (Quartz vein mining appears to be less common today, but was a big deal in the past; most of the well-known gold rushes went after quartz veins.) I'm told beryl (emerald, aquamarine, heliodor) is associated with tin ores.
Aside from this, it's worth noting that some metal ores can be passably pretty rocks in their own right; precious metal ores wouldn't be used that way since they're more valuable processed, but e.g. hematite (a major iron ore) can be either a glossy black stone, which was valued as a gem in Victorian times, or banded blood red/black. Azurite and malachite are well-known semi-precious gems that are also copper ores.
Anyway, could it be possible for a mine rich in either iron or copper to also produce some gemstones? I don't mean the big ones like diamonds, sapphires, emeralds or rubies but stones like peridot or turquoise or topaz.
Absolutely. In fact, turquoise, and a number of ornamental stones such as malachite and azurite, are dependent on large concentrations of copper (turquoise is a hydrated phosphate of copper and aluminum).
Turquoise was among the first gems to be mined, and many historic sites have been depleted, though some are still worked to this day. These are all small-scale operations, often seasonal owing to the limited scope and remoteness of the deposits. Most are worked by hand with little or no mechanization. However, turquoise is often recovered as a byproduct of large-scale copper mining operations, especially in the United States.
Arizona is currently the most important producer of turquoise by value. Several mines exist in the state, two of them famous for their unique colour and quality and considered the best in the industry: the Sleeping Beauty Mine in Globe ceased turquoise mining in August 2012. The mine chose to send all ore to the crusher and to concentrate on copper production due to the rising price of copper on the world market. The price of natural untreated Sleeping Beauty turquoise has risen dramatically since the mine's closing. The Kingman Mine as of 2015 still operates alongside a copper mine outside of the city. Other mines include the Blue Bird mine, Castle Dome, and Ithaca Peak, but they are mostly inactive due to the high cost of operations and federal regulations The Phelps Dodge Lavender Pit mine at Bisbee ceased operations in 1974 and never had a turquoise contractor. All Bisbee turquoise was "lunch pail" mined. It came out of the copper ore mine in miners' lunch pails. Morenci and Turquoise Peak are either inactive or depleted.