Given $\gamma^\mu$ in 1+3 dimensions with signature $(+,-,-,-)$, how can I obtain Dirac matrices in 1+1 dimensions expressed in terms of the $\gamma^\mu$?
The converse problem, constructing the 1+3 dim $\gamma^\mu$s (4 × 4 matrices) out of the 1+1 dim ones (2 × 2 matrices) is solved systematically here in WP .
The stated one is straightforward, since the standard Dirac rep 1+3 ones amount to just $$\gamma^0=\sigma^3\otimes I, \qquad \gamma^1=i\sigma^2\otimes \sigma^1, $$ hermitean and antihermitean, respectively.
You then observe that, in anticommuting these two, since the second hermitean tensor factors commute, these second factors make no difference whatsoever in satisfying the Clifford algebra, and may be dropped altogether, so that $$ \sigma ^3 \mapsto \Gamma^0, \qquad i\sigma^2 \mapsto \Gamma^1, $$ where I have designated the 1+1 dim $\gamma$s (2 × 2 matrices) as $\Gamma$s, not a little perversely, if only to stick to your original notation. Check the Clifford compliance thereof!