Can someone give an example of an ideal $I \subset R= \Bbb{Z}[x_1,...x_n]$ with $R /I \cong \Bbb{Q}$? Question in the title. I have never seen a quotient of $R$ by a maximal ideal $I$ that is an infinite field, so I would also be interested in the case that $R/I$ is any infinite field. If no such examples exist, I would also like to hear about the case where $R$ is a polynomial ring over $\Bbb{Z}$ in infinitely many variables.

You can't find such an example: a maximal ideal $\mathfrak m$ in $\mathbf Z[x_1,\dots,x_n]$ has a non-zero intersection with $\mathbf Z$, which is a prime ideal $p\mathbf Z$, hence the quotient $\mathbf Z[x_1,\dots,x_n]/\mathfrak m$ is a finitely generated algebra over the finite field $\mathbf F_p$, which has characteristic $p$.
The same argument is valid for polynomial rings with an infinite number of indeterminates.
