The first known contact with aliens in the Star Trek universe wasn’t with The Vulcans.
We all know it by now, The Vulcans saw Zefram Cochrane’s warp launch, diverted course, and made first contact with humanity on Earth. It led to the creation and founding of the United Federation of Planets and interstellar exploration many believed to be impossible. It was not, however, the first time humanity crossed paths with aliens, it wasn’t with the Vulcans.
It was with the Borg.
The events of the Vulcans making formal contact with humanity actually followed Lilly, coming face to face with The Borg. Lilly was Cochrane’s friend, confidant, and possible romantic partner and was injured during the opening moments of the film First Contact. She was discreetly taken to the USS Enterprise where she met and dealt with the Borg; making her the first person from the 21st century to make contact with an alien race, officially.
As a “first contact” only needs for two members of an alien race to meet in some form or capacity, Lilly’s brief involvement with the Borg makes her the first human to have official contact with an alien race.
Why “First Contact” still belongs to the Vulcans; technically
Lilly met the Borg before Cochrane met the Vulcans. That’s just a fact. The problem comes with the loose definition of “first contact” as seen by Starfleet. According to Starfleet Command Directive 010, first contact is an official meeting, not one of happenstance.
"Before engaging alien species in battle, any and all attempts to make first contact and achieve non-military resolution must be made."
Since Lilly was in fact engaging the Borg in battle, or at least attempting to avoid one, you could make a solid argument that her encounter wasn’t a first contact in the eyes of future Federation rule. This would establish Cochrane and The Vulcans as the first official meeting of two different species. Yet, by the basic definition of making contact with an alien race previously unknown to you, then yes, Lilly did make first contact with an alien race before anyone else in the history of the Star Trek universe.
It should also be noted because someone will bring it up, that while Vulcans were on Earth in the early 20th century, their presence was kept a secret. First contact needs two parties who are aware of one another. Lilly was aware of the Borg, Cochrane was aware of the Vulcans. The citizens of Carbon Creek were not aware of the Vulcans.