Is this line in Harry Potter and the Sorcerer's Stone a reference to Fantastic Beasts and Where to Find Them?

It is not a reference to Fantastic Beasts.

These lines are meant to be clues to Harry, Ron, and Hermione (and to us) that Hagrid’s interaction with this inconspicuous person with a dragon egg was actually something rather important.

In the movie (not the books), Hagrid is the kind of person that will divulge possibly important information –- and not realize it. In this case, Hagrid spilled the beans on how to get past a three headed dog. When questioned, Hagrid recalled how he had to prove himself to this stranger that he could handle the care and feeding of a dragon. He did so by telling the history of how Hagrid mastered the art of taming and caring for a three-headed dog.

By my interpretation, Hagrid did not know that he was being manipulated by someone who knew him personally, and knew that he had tamed a three headed dog. That someone also knew that this dog was being used as one of the guarding stages that protected the Stone. So Professor Quirrell (Voldemort) used the dragon egg as a ploy to entice Hagrid to 'spill the beans'.

Note that, more than once in the movie, Hagrid would divulge something important to Harry, and then say "I should not have told you that ... I should NOT have told you that". This motif was a key clue that Hargid told somebody something important (and possibly not even realize it).

Please note that Fantastic Beasts was written essentially as a follow on to the seven book series (and the eight movie series).

Neither. The movie "Fantastic Beasts" simply referenced a line in the previously written Potter book so as to tie them in to each other.

Don't forget; the whole Potterverse was meant to end with the final Harry Potter book. Rowling never had any intention of going beyond that.

It might have started out being a coincidence, since Fantastic Beasts wasn't yet planned, but I think one could argue that even something like these Harry Potter lines of dialog are being made to at least be a thematic link to Fantastic Beasts in tying both series more tightly together as companion pieces.

Some examples of connectivity:

Hagrid, like Newt, loves magical creatures to a fault (Spider debacle = "You never met a monster that you couldn't love") and Luna Lovegood, Newt's blood relative who, also like Newt, sees the good in things and notices things other do not, tends to be Harry Potter's animal/creature caretakers or introspective creature characters.There is also perhaps Sirius Black's running around as "Padfoot" and his friendship with Professor Lupin a werewolf, Professor Sprout, like Newt, A Huffelpuff (Mandrakes = Bowtruckles), Bill Weasley (raises dragons for a living), or Voldermort's fondness of Nagani...

What's curious about this is that the second film, 'The Crimes of Grindelwald' seems to be making much more effort to tie things together.

We already know the Elder Wand, one component of The Deathly Hallows, appears at the end of Fantastic Beasts, but from the three Crimes of Grindelwald trailers,

we also now know that Nicolas Flamel and The Philosopher's Stone are featured, along with a younger Dumbledore, Nagani, flashbacks to Newt's time at Hogwarts, and a lake/underwater sequence similar to those during The Triwizard Tournament Lake scenes in The Goblet of Fire. Paris is also one location and the Triward tournament included characters from an all girls French Magical school. Harry saves one of >! the girls, Bill >!Weasley marries her older sister, Fluer.

At the end of the day Fantastic Beasts is thematic to the idea of exploring literal magical creatures/beasts, to those that are repressed or cursed into magical creatures/beasts, transfiguration, and/or more importantly, the figurative 'beasts' that can live inside any person, which is relative to characters relationships and/or one's fondness of another, despite that at some point, some will be overcome with darkness and others will be forced to fight those they thought they cared for, causing a struggle from within (ie "the heart"). This relates to both series, even though Fantastic Beasts is exploring the idea much deeper.

Like Jon wrote in the comments, I think this is definitely an "easter egg" and easter eggs do serve as reference to link and create theme.