Is there a finitely presented group with infinite homology over $\mathbb{Q}$?

Thompson's group F is an example. It's finitely presented and, according to this paper of Ken Brown, the integral homology is free abelian of rank 2 in every positive dimension.

This answers the other part of your question, not answered by Thompson's group. For each $i\geq 3$ there is a finitely presented group $G_i$ with the property that $H_i(G_i\mathbb{Q})$ is infinite dimensional. The first such examples were due to Stallings ($i=3$ and Bieri $i>3$). Take a direct product of $i$ copies of the free group $F$ on two generators, and define $G_i$ to be the kernel of the homomorphism $F^i\rightarrow \mathbb{Z}$ that sends each of the $2i$ standard generators to $1\in \mathbb{Z}$. These are the Bieri-Stallings groups. For $i=2$ the analogous group is finitely generated but not finitely presented, and for $i=1$ it is free of infinite rank. These groups can be viewed as special cases of the Bestvina-Brady construction.