Can you choose to automatically succeed on a saving throw you make because of a spell you cast?
For instance, if I am trying to intimidate someone by casting earthquake centered on myself, do I also need to make the Dex save to avoid falling prone? Or am I not at risk?
There is no general rule that exempts casters from making saving throws for their own spells and, if a specific spell had such a rule, it would state it overtly.
However, Sorcerers can choose to take this useful metamagic ability:
Careful Spell
When you cast a spell that forces other creatures to make a saving throw, you can protect some of those creatures from the spell’s full force. To do so, you spend 1 sorcery point and choose a number of those creatures up to your Charisma modifier (minimum of one creature). A chosen creature automatically succeeds on its saving throw against the spell.
A Sorcerer could use Careful Spell to allow you to automatically succeed on a saving throw, from a spell they cast, and not fall prone.
The Earthquake spell has a number of ongoing effects; Careful Spell would not protect you from these effects on subsequent turns after it was initially cast.
They could, however, also use Careful Spell to protect you from other similiarly impressive spells (ie. Fireball etc.), many of which have no ongoing effects.
The rules text says quite specifically that you can protect 'some of those' [...] 'other creatures'. 'Other creatures' cannot mean you.
To achieve this effect, you'd therefore need a friendly sorcerer to cast a spell on you using Careful Spell; you wouldn't be able to use Careful Spell to protect yourself.
At level 2 School of Evocation wizards gain access to the feature Sculpt Spells. This works in a very similar way to Careful Spell for sorcerers but is limited to affecting evocation spells. Earthquake is an evocation spell, as are many other impressive spells, without the ongoing effects that Earthquake has.
However, this solution is less optimal than Careful Spell for your exact scenario, as Earthquake is not on the Wizard spell list (it's on the list for Clerics, Druids and Sorcerers), though lots of other evocation spells are. To cast Earthquake a wizard would therefore need to be multi-classed.
According to the Earthquake spell, not only would you need to make a DEX save to avoid falling prone, but you'd also need to make a CON save just to maintain concentration on the spell!
Each creature on the ground that is concentrating must make a Constitution saving throw. On a failed save, the creature's concentration is broken.
As a general rule: Unless a spell specifically says it has no effect on the caster, you will not get an automatic save without a special class feature or Feat that provides it.
The spell Earthquake states:
You create a seismic disturbance at a point on the ground that you can see within range
If you are within the spell range, it does affect you, a way to ''auto-succeed'' would be to not touch the ground, such a thing can be achieved with the spell Fly (cast on you by an ally) or a magic item such as Wings of Flying.
You could also cast it somewhere that will not affect you.
You can't normally auto-suceed on a saving throw and not even choose to fail it. This question might help you: Can you choose to fail a saving throw?
