Let $G$ be a group containing a monoid $M$ that spans $G$ as a group. Is it possible to have a proper quotient $\varphi \colon G \to Q$ of $G$ such that the restriction of $\varphi$ to $M$ is injective?

More specifically I'm interested in the following: if $M$ is an Ore monoid (cancellative and admitting least common right multiples) then it embeds into its group of right fractions $Q$. There is also a universal group $G$ through which any map from $M$ to a group factors (it has presentation $\langle M \mid m \cdot n = (mn) \text{ for }m,n \in M\rangle$). So $Q$ is a quotient of $G$. Can it be a proper quotient?

In other words: $Q$ is by definition universal among the groups $\iota \colon M \to H$ subject to the condition that $H = \iota(M) \cdot \iota(M)^{-1}$. Is it nontheless universal among all groups?