There are several confusions in your post.

**1.** Automorphic forms for the group $\mathrm{SL}_2(\mathbb{Z})$ have level $q=1$ by definition. You probably wanted to talk about newforms for the Hecke congruence subgroup $\Gamma_0(q)$.

**2.** Good only treated the case of $q=1$. The bound you claim was only proved recently (with an unspecified $A>0$) by [Booker-Milinovich-Ng (2009)][1] and [Aggarwal (2020)][2].

**3.** It is not known that $A=1/4+\varepsilon$ is admissible in the bound you claim. One can take $A=1/4+\varepsilon$ if at the same time the exponent of $|t|+1$ is enlarged to $1/2+\varepsilon$; this is the convexity bound.

**4.** As far as I know, that best hybrid subconvexity bound for the family at hand is due to [Wu (2022)][3]:
$$L(f,1/2+it)\ll_\varepsilon (q(|t|+1)^2)^{1/4-1/224+\varepsilon}.$$ If the nebentypus of $f$ is fixed, then $1/224$ can be improved to $25/1536$.


  [1]: https://doi.org/10.1016/j.aim.2018.10.037
  [2]: https://doi.org/10.1016/j.jnt.2019.07.018
  [3]: https://doi.org/10.1112/mtk.12147