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In this article we study the equations $x^4+dy^2=z^p$ and $x^2+dy^6=z^p$ forpositive square-free values of $d$. A Frey curve over $\mathbb{Q}(\sqrt{-d})$is attached to each primitive solution, which happens to be a$\mathbb{Q}$-curve. Our main result is the construction of a Hecke character$\chi$ satisfying that the Frey elliptic curve representation twisted by $\chi$extends to $\text{Gal}_\mathbb{Q}$, therefore (by Serre's conjectures)corresponds to a newform in $S_2(n,\varepsilon)$ for explicit values of $n$ and$\varepsilon$. Following some well known results and elimination techniques(together with some improvements) it provides a systematic procedure to studysolutions of the above equations and allows us to prove non-existence ofnon-trivial primitive solutions for large values of $p$ of both equations fornew values of $d$.

$\mathbb{Q}$-curves, Hecke characters and some Diophantine equations