Does Exploit Weakness work with quake and/or poison

Just out of curiosity, what two characters have Exploit Weak. and either one of those?

Poison is not an attack in the standard sense so EW doesn't apply there. Quake is a close combat action which can only work as stated and EW doesn't work here either. This is because EW is a close combat action and you aren't allowed to make two non-free actions with one character. Any power which affords you a "close combat action as a free action" will work with EW (such as Charge).

Mr Bones has poison and EW.
KC and JSA Shazam! have Quake and EW. Well, Magog does too but doesn't have them on the same click.

First one: Quake and Exploit Weakness

QUAKE (optional): Give this character a close combat action. 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 characters adjacent to this character. Each character that takes damage suffers knockback.

EXPLOIT WEAKNESS (optional): Give this character a close combat action. Damage from this attack cannot be reduced by any power that reduces damage dealt.

Both require a close combat action. So the answer is no. You can do either with Charge, which allows a close combat action.

Your second, Poison and Exploit Weakness

POISON (optional): At the beginning of your turn, each opposing character adjacent to this character is dealt 1 damage.

Since this is not an attack, EW does not function. EW requires a close combat action and Poison is not an action.
So, this one is no as well