After a translation in the variable $x$ we can assume that the point on the quartic is $(0,q)$. Then plugging in this point into the equation, we get an equation of the shape:
$$ y^2 = Ax^4+Bx^3+Cx^2+Dx+q^2\ .$$
For the simplicity of the following formula, let us write $Q=2q$.

The following transformation is given in [Ian Conell, Elliptic Curves Hanbook, page 105, Quartic to Weierstrass] . (Wonderful exposition.)
$$ x = ( Q(v+q)+Du )/u^2\ ,$$
$$ y = ( Q^2(v+q) + Q(Cu^2+Du) - D^2u^2/Q)/u^3\ .$$

(For a sage example with "simpler" coeficients: https://ask.sagemath.org/question/36637/change-of-variable-from-hyperellictic-curve-to-weierstrass-form/

(This is of course a community wiki post... All karma goes to all beautiful books and people in this field.)