Let me point out a somewhat different answer. Rbega explicitly mentions
$$
Q = \left(\frac1{z^3} + \frac1{z^2}\right)\ (dz)^2
$$
as an example of the sort of meromorphic quadratic differential that is of interest, and this is not at all covered by dalakov's answer.

Consider the more general problem of asking when there is a normal form (possibly with parameters) for differentials of the form
$$
Q = \frac{h(z)\ (dz)^l}{z^k}\ ,
$$
where $k$ and $l$ are fixed positive integers and $h$ is holomorphic near $z=0$ with $h(0)\not=0$.

Now, when $k$ is not a multiple of $l$, there exists a local coordinate $w$ centered on $z=0$ such that
$$
Q = \frac{h(z)\ (dz)^l}{z^k} = \frac{(dw)^l}{w^k}\ ,
$$
so all of these differentials are locally equivalent.

However, when $k=ml$ for some integer $m>0$, the story is quite different. For example, it is clear that the 'residue'
$$
R_{l,l}\left(\frac{h(z)\ (dz)^l}{z^l}\right) = h(0)
$$
is well-defined, independent of the choice of 0-centered local coordinate $z$. Moreover, in this case, there exists a coordinate $w$ centered on $z=0$ such that
$$
\frac{h(z)\ (dz)^l}{z^l} = \frac{h(0)\ (dw)^l}{w^l},
$$
so the 'residue' $h(0)$ is the only invariant in the case $k=l$.

Now, in all the cases in which $k=ml$, there is such an invariant. For example, when $k=2l$, one has the 'nonlinear residue'
$$
R_{l,2l}\left(\frac{h(z)\ (dz)^l}{z^{2l}}\right) = h(0)\left(\frac{h'(0)}{h(0)}\right)^l,
$$
which is independent of choice of coordinates. One also has
$$
R_{l,3l}\left(\frac{h(z)\ (dz)^l}{z^{3l}}\right)
= h(0)\left(l\ \frac{h''(0)}{h(0)}-(l{-}1)\ \left(\frac{h'(0)}{h(0)}\right)^2\right)^l,
$$
and
$$
R_{l,4l}\left(\frac{h(z)\ (dz)^l}{z^{4l}}\right)
= h(0)\left(l^2\ \frac{h'''(0)}{h(0)}-3l(l{-}1)\frac{h'(0)h''(0)}{h(0)^2}
+(2l{-}1)(l{-}1)\ \frac{h'(0)^3}{h(0)^3}\right)^l,
$$
and so forth.

The general rule is that, in these cases, one can write
$$
Q = \frac{h(z)\ (dz)^l}{z^{ml}} = \left(\frac{\bigl(h(z)\bigr)^{1/l}\ dz}{z^m}\right)^l
= \omega^l\ ,
$$
where $\omega$ is a meromorphic $1$-form, well defined near $z=0$ up to multiplication by an $l$-th root of unity. The $l$-th power of the usual residue of $\omega$ at $z=0$ then provides an invariant that works out to be a rational expression in the coefficients of the power series of $h$, as in the cases noted above.

Finally, when $m>1$ and $h(0)\not=0$, it can be shown that there exists a local coordinate $w$ centered on $z=0$ such that
$$
\frac{h(z)\ (dz)^l}{z^{ml}} = \left(\frac{(1 + a\ w^{m-1})\ dw }{w^m}\right)^l\ .
$$
In this case, the 'residue' is (up to a universal constant multiple) simply $a^l$. An alternative normal form is
$$
\frac{h(z)\ (dz)^l}{z^{ml}} = \frac{(1 + b\ \zeta^{m-1})\ (d\zeta)^l}{\zeta^{ml}}
$$
for some constant $b$, and the nonlinear residue in this case is (a universal constant multiple of) $b^l$.