There's not a lot to go on in official 5E materials, but there's a few clues in the monster description which we can extrapolate from. Note that since the default setting in 5E is Forgotten Realms, what follows is Forgotten Realms lore. Unicorns in other settings might be very different.
To start, these are regional effects that might apply to a unicorn's domain:
- Open flames of a nonmagical nature are extinguished within the unicorn’s domain
- Creatures native to the unicorn’s domain have an easier time hiding; they have advantage on all Dexterity (Stealth) checks made to hide.
- When a good-aligned creature casts a spell or uses a magical effect that causes another good-aligned creature to regain hit points, the target regains the maximum number of hit points possible for the spell or effect.
- Curses affecting any good-aligned creature are suppressed.
And it says "If the unicorn dies, these effects end immediately." I think it's therefore reasonable to expect these things to be manifestations of the unicorn's wants.
- they don't like fire. This might put them at odds with fire as an elemental force, and opposed to, say, any cults of fire or fire deities.
- hiding and stealth are valued as a way to stay safe, and healing and protection of good creatures is a priority.
Then, also from the description:
Unicorns allow good-hearted creatures to enter their woods to hunt or gather food, but they hold evil ever at bay.
So, again, protection of the good — and holding evil ever at bay. And not just passively, because:
Foul-hearted creatures seldom leave a unicorn’s domain alive.
On maybe a more local, specific level:
Most unicorns protect a bounded realm such as an enchanted forest. However, the gods sometimes send a unicorn to guard sacred artifacts or protect specific creatures.
So, an individual, specific unicorn may want a particular location, artifact, or creature protected. That would be very campaign-specific, of course (not necessarily a bad thing).
Under a unicorn’s protection, creatures feel safe from the threat of encroaching civilization and the insidious spread of evil.
Again, keeping the spread of evil at bay — and also, stopping "encroaching civilization".
And finally from the monster description:
A paladin astride a unicorn is a sign of the gods’ direct intervention in the affairs of the mortal realm. It is a holy alliance made to cleave the heads from demons and banish devils back to the Nine Hells.
... we've got a specific remit to fight demons and devils.
That's almost all there is to go on, except the note in Sword Coast Adventurer's Guide that:
Mielikki’s symbol is a unicorn, which prompts some to think of her as such and conflate her with Lurue, Queen of the Unicorns and the actual goddess of their kind.
Unfortunately, as far as I can tell, there is no official 5E lore relating to Lurue, but one might be able to find some from older editions. One tiny tidbit comes from the Knight of the Order character background in Sword Coast Adventurers' Guide:
The Knights of the Unicorn [...] On a lark, they took the unicorn goddess Lurue as their mascot and went on various adventures for fun. The reality of the dangers they faced eventually sank in, as did Lurue’s tenets. [...] The Knights of the Unicorn are chivalric adventurers who follow romantic ideals: life is to be relished and lived with laughter, quests should be taken on a dare, impossible dreams should be pursued for the sheer wonder of their completion, and everyone should be praised for their strengths and comforted in their weaknesses.
We don't know which of these are actually Lurue's tenets in 5E, but this paragraph is actually directly lifted word-for-word from the description of Lurue's dogma in the 3rd edition (3.0, pre 3.5) book Faiths and Pantheons (which in turn seems to have been adapted from a similar one in AD&D 2e Powers and Pantheons). So that seems plausibly canonical.
A warlock following that particular happy-go-lucky knight errant character trope would be quite different from warlocks as often played, and also quite different from the stereotypical follower of a forest god.
Also as suggested in the Forgotten Realms wiki, both of the older-edition sources note that Lurue is an enemy of Malar, the beast-god of lycanthropy and is also opposed to the Red Wizards of Thay — so, there are some other specific sources of evil incarnate your unicorn patron might have particular concerns about.