Exploit can be used with Quake correct?

Nope. Both require a close combat action to activate.

No. They cannot be used together.

QUAKE
Give this character a close combat action; until the attack has been resolved, this character’s damage value becomes 2 if it is greater than 2. Make a single attack roll and compare the result to the defense values of all opposing adjacent characters. Each character that takes damage from this attack is knocked back.

EXPLOIT WEAKNESS
Give this character a close combat action. Damage from the attack is penetrating damage.

Both Powers require a close combat action to activate. You can only assign one non free action to a figure in a turn, so you cannot use both.

Tarnish