Is $x^TAy = y^TAx$ for any matrix $A$? I know that $x^TAy = y^TAx$ is true for symmetric quadratic matrices, but, it is true for non symmetric quadratic matrices?

No. Let $e_i$ denotes the vector with a $1$ at the $i$-th position and zeros elsewhere. If $A$ is not symmetric, then $a_{ij}\
e a_{ji}$ for some $i\
e j$, but then $e_i^TAe_j=a_{ij}\
e a_{ji}=e_j^TAe_i$.