The chances are very very slim

I actually think the chances are pretty good. They are both Ravenclaws and likely overlapped a few years while they were at Hogwarts, though I can't be certain, because Quirrell's year of birth is unknown. Furthermore, Hogwarts Mystery has Lockhart visiting Hogwarts a few times at a time Quirrell would have been teaching Muggle Studies.

Yes and here's why: Lockhart was born 3 years before Quirrell, thus they had 4 years at Hogwarts together. Both were in Ravenclaw and Quirrell decided to try out for the Ravenclaw's Quidditch team only to realize his fear of possibly falling off a broom. It's likely after Hogwarts Quirrell and Lockhart kept in contact over the summers. If this is the case, Quirrell likely told Lockhart he had been hired by- none other than- Albus Dumbledore as the Defense Against The Dark Arts teacher. After Quirrell's death, Dumbledore could've hired Lockhart knowing how close they were, and when explaining Quirrell's death, it could have caused even more panic in Lockhart's mind.

^^^ that’s cool