Here is the wording of the BFC: Proximity Mines

At the end of a turn, if any character is in the same square as an uncarried object or in a square adjacent to an uncarried object, roll one six-sided die. On a result of 5 or 6, deal 1 damage (that ignores all effects that reduce damage) to all characters in the same square as or adjacent to the object, and then remove the object from the game.

My question is, how does the object work with elevated terrain. Say for instance the object is on the ground next to a character on elevated terrain. Would that character be affected if the proximity mine detonates?

No. Elevated terrain is not considered adjacent to non-elevated terrain.

As Quebbster said, but with further clarification:

If the object is on the ground, next to a wall, and there is a character on top of the wall (on elevated terrain), OR, if the object is on the elevated terrain, and the character is on the ground, they are not adjacent.

It doesn't say "in the same square or an adjacent square," it says "in the same square or adjacent to the object". HUGE difference.