So I am very new to playing Talisman and while playing with my son we came across a disagreement.

When two characters meet in the city/village/tavern/chapel can a battle happen?

Yes. The character that lands there can choose to encounter the space OR the character who is already there.

This is important for two spaces, The Chapel and The Graveyard. Usually if you are evil and you go to The Chapel you lose a life. The same goes for The Graveyard if you are good. If a character is on either of these spaces and you are not of the right alignment (good for The Chapel / evil for The Graveyard) you can attack the character and not lose a life for landing on that space.

And desert.

Oh, but the exception to this rule: you can't encounter characters in the Inner Region except on the Plain of Peril and Valley of Fire.