In this paper, we study Diophantine exponents $w_n$ and $w_n^{∗}$ for Laurent series over a finite field. Especially, we deal with the case $n = 2$, that is, quadratic approximation. We first show that the range of the function $w_2 − w_2^{*}$ is exactly the closed interval $[0,1]$. Next, we estimate an upper bound of the exponent $w_2$ of continued fractions with low complexity partial quotients.