a) There are old results which directly imply the existence of such groups:
(1) Boone Higman 1972: every f.g. group with solvable word problem embeds into a simple subgroup of a finitely presented group.
(2) Every countable group with solvable word problem embeds into a f.g. group with solvable word problem (reference? the original HNN construction directly works since it consists of explicit amalgams — alternatively here Ph. Hall produced in the 50s an explicit 3-generated metabelian group with solvable word problem, with copies of $\mathbf{Q}$).
b) A more explicit example is the group $\tilde{T}$ obtained as the set of self-homeomorphisms of $\mathbf{R}/\mathbf{Z}$ commuting with $\sigma:n\mapsto n+1$, that are piecewise affine with dyadic slopes and breakpoints.
The center of $\tilde{T}$ is the infinite cyclic group $\langle\sigma\rangle$ and the quotient is naturally identified with Thompson's group $T$, which is a finitely presented simple group. The normal proper subgroups of $\tilde{T}$ are precisely the subgroups of $\langle\sigma\rangle$.
If $Q$ is any copy of $\mathbf{Q}$ in $\tilde{T}$, it follows that the image of $Q$ in $T$ is non-central (since $\mathbf{Q}$ has no nontrivial cyclic quotient). Hence the same holds in every nontrivial quotient of $\tilde{T}$.
Finally, that $\tilde{T}$ contains a copy of $\mathbf{Q}$ (and even continuum many such copies) is an original observation of Belk, Matucci, Hyde (arXiv).
(The other answer is closely related as it refers to a more complicated finitely presented simple group containing $\tilde{T}$. The group $\tilde{T}$ itself is not finitely presented but has few enough normal subgroups for the condition to hold.)