No, it is not possible because the induced metric on any submanifold $N$ of the Euclidean space $E^3$ is necessarily positively defined, whereas the metric on $1+1$ Minkowski space is indefinite.

The reason is trivial: The scalar product $<u,v>$ of two vectors $u,v$ in $N$ is, by definition, the scalar product in $E^3$ of these vectors viewed as vectors in $E^3$, so that $<u,u> \geq 0$ -- where $<u,u>=0$ implies $u=0$ -- in any cases, contrarily to what may happen in Lorentzian manifolds.

No, it is not possible because the induced metric on any submanifold $N$ of the Euclidean space $E^3$ is necessarily positively defined, whereas the metric on $1+1$ Minkowski space is indefinite.

The reason is trivial: The scalar product $\langle u,v\rangle $ of two vectors $u,v$ in $N$ is, by definition, the scalar product in $E^3$ of these vectors viewed as vectors in $E^3$, so that $\langle u,u\rangle \geq 0$ -- where $\langle u,u\rangle=0$ implies $u=0$ -- in any cases, contrarily to what may happen in Lorentzian manifolds.