An elliptic curve $E$ defined over $\bar{\mathbf{Q}}$ is called a $\mathbf{Q}$-curve, if $E$ and $E^\sigma$ are isogenous over $\bar{\mathbf{Q}}$ for any $\sigma$ in $\mathrm{Gal}(\bar{\mathbf{Q}}/\mathbf{Q})$.
For a real quadratic field $K$ and a prime number $p$, we consider a $\mathbf{Q}$-curve $E$ with the following properties:
1) $E$ is defined over $K$,
2) $E$ has everywhere good reduction over $K$,
3) there exists a $p$-isogeny between $E$ and its conjugate $E^\sigma$.
In this paper, a method to construct such a $\mathbf{Q}$-curve $E$ for some $p$ will be given.