Talk to Iris near the entrance to Dragonspine mountain, and ask about the blue crystals. You will get a quest called "In the Mountains".
Start by unlocking the Frostbearing Tree. You unlock it by using the surrounding scarlet quartz and attack the crystal in the tree, one at a time, for 4 times.
Head to the location shown above, south of the statue of the seven. You will see an entrance to an ice cave inside the mountain, next to a closed circle door.
Head ALL the way to the bottom and do the time trial challenge. It is highly encouraged that you use fire element characters, or at the very least, guide 1 or 2 of the heat seelies to its place before starting the trial. There will be 3 waves of enemies, and a lot of frost archers as well as falling crystals.
Once you are done with the trial, destroy the blue crystal, similar to unlocking the Frostbearing Tree. 2 of the scarlet quartz are near the time trial device. The other 2 are behind the blue crystal area. One of them is right next to a broken golem, inside some breakable rocks, the other is the one with a path that leads up.
Don't forget to also pick up the Crimson Agate that's above the blue crystal before you leave.
Head to the colosseum-like area in Wyrmrest Valley. It's easier to reach this area by going to the teleport waypoint from north of this location. Activate the device, and the warming seelie will guide you to the pattern of pillar activation. It starts from the bottom left, then upper right, lower right, upper left, then top middle. After you finish it properly, a treasure chest with a black mist will appear. Open it, and the surrounding enemies will wake, forcing you into a fight.
The fight starts with 2 regular ruin guards, except the weather changes to a snowstorm/blizzard, increasing the rate of sheer cold buildup. Around the area, outside of the ice pillars, are 4 heating devices. They light up one at a time. You have to pay attention and see which one is currently lit, so you can lower the sheer cold to manage the fight.
Once you take down the 2 Ruin Guards, the Ruin Grader will awake. When it attacks, it will briefly expose its core on the legs and eye. Attack it when it's open to disable it for a few seconds.
After you defeat the Ruin Grader, the entrance to the lower level is open. You will have to destroy the blue crystal again. 3 of the scarlet quartz are pretty close to it, but the 4th one requires you to guide 2 warming seelies to their places to unlock a gated area. Once you break the blue crystal, 2 low hp Ruin Graders will awake.
In the gated area, there is some ice wall that you can break by using the scarlet quartz. There will be a warming seelie that guides you to the exit, as well as showing you the crimson agate above the ice shown above. Follow the warming seelie outside and get some crystals along the way. When you reach the exit, head to the west to activate the teleport waypoint.
Center of the Mountain
Across the Statue of the Seven, there is a path inside into the middle of the mountain. Go inside the fairly linear path, and later climb up until you reach an exit.
After exiting up, you will see a big monolith and floating rocks. There are 3 blue crystals that you have to destroy.
1st scarlet quartz
You can find the first scarlet quartz close to the teleport waypoint, just slightly higher, right next to the broken down ruin guard. In case it's not there, or not spawning, you can do the next ones instead.
2nd scarlet quartz
Follow the circle broken stairway/path all the way to the top, to find your first agate near a frost lawachurl. After you defeat the lawachurl, there will be a scarlet quartz and a warming seelie. Start with the scarlet quartz first, then go back to the area to follow the warming seelie to the top.
3rd scarlet quartz
After activating the Anemo pillar, you will be guided to the other side. The second/third scarlet quartz is next to the fire, close to the wall. You can still continue to go up for some treasures and crimson agates.
Note: It might be bugged at the time of this writing, because this scarlet quartz will spawn again after you destroy it, making it the second and third scarlet quartz required. Others have noted that the first scarlet quartz is at the bottom, near the teleport waypoint.