I want to know if what I am doing is right.
I know that if $f$ is an automorphism, so $f(2)=2\implies f(\sqrt{2})^{2}=2.$ So, $f(\sqrt{2})=\pm \sqrt{2}$. In the same way, $f(3)=3\implies f(\sqrt{3})^{2}=3\implies f(\sqrt{3})=\pm \sqrt{3}.$
So, setting
$$f_{1}(\sqrt{2})=\sqrt{2},\;f_{1}(\sqrt{3})=\sqrt{3}\; ; \\ \quad f_{2}(\sqrt{2})=\sqrt{2},\;f_{2}(\sqrt{3})=-\sqrt{3}\; ; \\ \quad f_{3}(\sqrt{2})=-\sqrt{2},\; f_{3}(\sqrt{3})=\sqrt{3}\; ; \\ f_{4}(\sqrt{2})=-\sqrt{2},\;f_{4}(\sqrt{3})=-\sqrt{3};$$
we have $\textrm{Aut}_{\mathbb{Q}}(\Bbb{Q}(\sqrt{2},\sqrt{3}))=(f_{1},f_{2},f_{3},f_{4})\simeq \Bbb{Z}_{4}.$
Am I right?