Your curve can be written as $Y^2=X^4+X^3-2,$ where $Y=4n$ and $X=q^2.$ This Diophantine equation satisfies Runge's condition, so this is relatively easy to handle and one obtains that there are only finitely many integral solutions (see [Poulakis-Quartic][1]). You may also consider it as a genus 1 curve and there are techniques to determine all integral points on such curves (see [Tzanakis-Quartic][2]).


  [1]: https://link.springer.com/article/10.1007/s000170050053
  [2]: http://www.math.uoc.gr/%7Etzanakis/Papers/Ellqua-v2.pdf