How did Voldemort (within Quirrell) come to know that the Philosopher's Stone was in Harry's pocket?

Here is an answer from Sister Site Science Fiction & Fantasy:

There could be two possible answers to this:

1) The possible answer could be Legilimency. Voldemort was highly skilled in the art of Legilimency.

Here is a quote of Harry Potter Wiki:

"Voldemort has used Legilimency extensively, both wandlessly and nonverbally, to enter the minds of those he wished to interrogate."

The quote can be found here, under History, third paragraph, first line.

This shows that Voldemort knew what he was doing because of his extensive knowledge, and this quote shows his skill:

"You flatter me," said Dumbledore calmly. "Voldemort had powers I will never have."

Voldemort was a very intelligent wizard.

To get to your question, here is another quote found on the Harry Potter Wiki:

"If a target is not skilled in Occlumency, a Legilimens will be able to detect if the person is lying, as well as delve into their thoughts, emotions, and memories."

The quote can be found here, under Methods, first paragraph, second line.

Harry was not even aware of Occlumency and Legilimency, so obviously he would be "bad" at it. This was evidently shown during, and after the fifth book, during, and after his lessons with Snape. Voldemort could have detected, through Quirrell that Harry was lying, and furthermore "delved into his thoughts", and saw that has was thinking about the stone in his pocket.

2) The other possible reason could be because of the fact that Voldemort had a mental connection to Harry's mind, which may have let him more easily read what Harry was thinking.

A third possibility, not mentioned above, is that Voldemort is simply good at reading people. While I'm not sure how much made it into the movies, but the original narrative of Deathly Hallows indicates that, even before he was introduced to magic, Tim Riddle was very talented at reading and manipulating his fellow orphans. It may have been Legilimency when he was shown manipulating Hagrid in Chamber of Secrets, but it could go either way. Tom was a sixth year student, so could have learned the discipline (in class or on his own). Or he could have just been using the same tricks from earlier in his life.

Reading a scared and generally guileless eleven year old would have been easy for him.

Also, it was not clear what kind of magic he could do in his wraith state, but it was at least in some way limited. So that is a point against actual Legilimency .