MOST of the gems would either be the same or have close analogs. Depending on chemical abundance, you may have different combinations. Gem colours are usually impurities -- ruby is chromium.
You may get gem quality stones of things that normally don't form gems here. E.g. natural cubic zirconia has been found, but only microscopically. There are some oddball gems that are uncommon on earth that may be common there.
I would certain expect all the stones based on quartz and corundum (silicon dioxide, and aluminum oxide) Amethyst and kin, sapphire and kin to be present.
Suggest reading up on Wiki on the particular gems you have in mind.
gives you a good start. Many are not observed to be gems, but it gives you a lot to play with.
If you want different ones, you have to do some worldbuilding geology -- see if you can come up with a plausible way to come up with different crystal structures. This may be accomplished by having radically different concentrations of metals, or perhaps something different in the way of solvents. (Could you get different crystals on a world that ran 10% ammonia for it's liquid phase...)
Some stones can be modified by heat. Heating them will change the colour. The colour of amethysts is influenced by radiation
Another way for different gems are biological origins. In Piper's Fuzzy series, the planet Zarathustra has "sun stones". Apparently if a certain jellyfish died, and was embedded the right way, it turned into a stone that was thermo-luminescent.
Petrified wood is another example: different parts were replaced at different times, and hence have differing colours.
Look at microscopic pictures of radiolarians. Imagine such a critter much larger. Imagine them with crystaline sapphire skeletons.
If you want to have fun, put down gems that cannot be grown by natural processes, but instead have to be built by a molecular assembler. (What kind of gems could you get with a fractal array of copper inside? Or gems that have books coded into them. Gems that are in effect 3d hologram encoded picture catalogs.) You can justify this with a culture that is trying to leave a really long time record. 5 carat holographic diamonds, made by the trillions. With layers of information in them that required more and more sophisticated tools.
If you are going to make artificial gems, look at making them with unusual symmetries. square, triangular and hexagonal symmetry is common. Mirror, pentagonal is more rare. Heptagonal (7 way) is unknown.