I found a way to... ahem... obtain the animated series, so here goes:
Pretty much the only similarity between the two versions of how they meet is that both are set in medieval France and involve the two characters being imprisoned in a castle.
In the animated series, Valérian is a newbie space-time agent sent on his first mission as an "observer" to 10th century France. There, Laureline is a juggler and acrobat travelling with a troupe of entertainers. A local lord takes a fancy to her and when she declines his advances, he imprisons her, along with Valérian who made a snarky comment when witnessing this.
The two escape their cell, Valérian flees towards his ship (through a string of incredibly badly written and choreographed fighting scenes), Laureline follows him, and when he is knocked unconscious, she drags him into the ship, which takes off. Then they return to the future and notice that, due to this interference with the past, the Earth is gone.
In the comic, Valérian is an elite space-time agent sent to 11th century France to hunt down a high-level technocrat who stole a time machine with unknown purpose. He follows the man into a cursed forest. There, he makes camp and, when he wakes the next morning, finds himself trapped inside a giant leaf. Laureline appears, says she'd followed him the day before and is familiar with the cursed forest. She frees him, offers her services as a guide and tells Valérian that a sorcerer living in a nearby castle should be able to help him find the fugitive.
Arriving at the castle, the two are overpowered and learn that the fugitive has taken over the castle after learning magic from the sorcerer. He has hired some henchmen and turned people into mythical creatures, planning to go to the future and use these to take over Galaxity, since most people there are decadent and easily terrified. Of course, the two heroes manage to escape and foil his plan. The disappearance of Earth is the main later story arc of the comic and completely unrelated to this origin story.