Faction Mounts are obtained by turning in faction coins to a city that has the faction mount you want.
At level 10 you can talk to an npc in the city you want to gain faction for, choose the dialogue option to fight for that city, and wear the diplomacy necklace given to you to collect the faction coins from mobs that give you xp. They drop randomly and are automatically added to your inventory.
You will gain copper, silver, and gold faction coins depending on your level and they will be turned in for faction points for the city. You can gain platinum faction coins from boss mobs who are not grey to you and whose names are all in CAPS. These are what you use to pay for the mount. You can also get ancient platinum coins through various means such as crafting or diplomacy. These can be exchanged for platinum coins of any city faction.
1 copper coin = 1 city faction
1 silver coin = 10 city faction
gold coin = 100 city faction
The level 30 mount costs 50 platinum faction coins and requires 1000 city faction, the level 50 mount is 150 platinum faction coins, or 100 platinum faction coins if you turn in your level 30 mount, and 18,000 city faction.
WARNING - Be aware you will lose faction from opposing cities and may become Kill on Sight to those cities. You will also lose citizen faction which, when it gets below a certain point, you will be unable to acquire quests or use vendors from that city. If you want to remove the KoS status, you need to raise your faction again with the cities that hate you by killing mobs, diplomacy, or crafting work orders that give positive faction for those cities.