$I=\langle x^2+2y^2-3, x^2+xy+y^2-3\rangle$ in $\Bbb Q[x,y]$ how do I find the Grobner basis for $I \cap \Bbb Q[x]$ and $I\cap\Bbb Q[y]$?

Question

Given the ideal $$I=\langle x^2+2y^2-3, x^2+xy+y^2-3\rangle$$ in $ ℚ[x,y]$, how do I find the Grobner basis for $I \cap \Bbb Q[x]$ and $I\cap\Bbb Q[y]$?

I have learned to find a Grobner basis using the Buchberger algorithm but it was with simpler polynomials and I don't know how to do it in the case of $I \cap \Bbb Q[x]$ and $I\cap\Bbb Q[y]$.

Any help would be great.