What different opening strategy exist when playing Quatro?
I don't know that you're going to do much better than random.

Clearly, when you are not the first placing player your piece choice doesn't matter.  All the pieces are equivalent (the same number of potential matches will remain after play).

If you are the first placing player, then there are only 2 distinct opening moves.

Placing any of the pieces on:


a "corner*"

the "inner" pieces are functionally equivalent to outer corners on the first move

a "wall" but not a "corner"


This is due the board's symmetry, and the lack of piece ownership, coupled with the equivalence of the piece "types."

None of these moves deny any potential win condition, clearly.

So, the first choice that has any impact on the outcome of the game can't occur until the second piece choice.

Here, the first placing player can choose a piece with 3, 2, 1, or no commonalities with one just placed.

This means that by the end of the first placing player's turn, there can be only 8 functionally distinct boards.  Not a lot to choose from in terms of strategy.  In comparison, Chess has 20 at this point and Checkers has 7.

A bit less fairly, this analysis suggests that given perfect play the first placing player will always win or force a draw.

*I apologize for the paraphrasing but I haven't actually played this game personally, only read about it.