How do I find Meridia's Beacon in Skyrim?

Please see Arremer's answer which is accepted. But if you still have problems then follow along:

I started the quest "The Break of Dawn", but when I set the quest as active it won't give me a quest marker.

There are two ways to start the Break of Dawn quest. The required level for this quest is 12.

1) Either you find Meridia's beacon while questing. It does spawn randomly.

2) The other option is to visit Meridia's statue, in which case she asks you to find the beacon in some camp. IF this is the case then the quest marker activates as long as your quest journal has that quest active. In this case Clairvoyance should help as long as this quest is active in your journal and no other quests are active.

If you used console commands advlevelor setlevel to get to the level 12 requirement for this quest, it will not start at all. To correct this, use player.setlevel 11 to get to level 11 and then level up normally or use incPCS to level up some of other skills to level up to 12. Source

Apart from this I have not encountered any bugs related to this quest.

I don't think the existing answers actually address what is likely happening here.

Generally speaking, for any of the randomly spawned quests (especially radiant quests), they can spawn anywhere that allows for it. For both Skyrim and Solstheim, the quest markers work fairly well, however, sometimes they will not span the two correctly. For example, a quest that spawns in Solstheim whose destination is Skyrim may not show a quest marker until you return to Skyrim.

This situation is exacerbated by installing Mods that add new lands which have destinations that support radiant quest destinations, for example Falskaar or Beyond Skyrim: Bruma. In the case where the destination is not on your current "map", the quest marker will not appear. Once you travel to that "map", the quest marker will appear, and you can complete the quest.