Let $\rho_p : \mbox{Gal}(\overline{\mathbb{Q}}_p / {\mathbb{Q}_p}) \to \mbox{GL}_n(\mathbb{Q}_p)$ be a de Rham $p$adic representation. Can one find a representation $\rho : \mbox{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mbox{GL}_n(\mathbb{Q}_p)$ such that $\rho$ is geometric (in the sense of FontaineMazur) and such that the restriction of $\rho$ to $\mbox{Gal}(\overline{\mathbb{Q}}_p / {\mathbb{Q}_p})$ is $\rho_p$ ?

No. There are uncountably many unramified representations from the local Galois group to $\mathbb{Q}^{\times}_p$, since Frobenius can be sent to anything in $\mathbb{Z}^{\times}_p$. However, there are only countably many global representations of this form, since, by class field theory, they all factor through the Galois group of a (finite) cyclotomic field. The general picture is pretty much the same  the local Galois representations form large $p$adic analytic families, yet, assuming the FontaineMazur conjecture, there are only countably many geometric representations. 

