On explicit embeddings of $\mathbb{Q}$ into finitely presented groups

Explicit embeddings of the group $\mathbb{Q}$ into a finitely presented group $\mathcal{Q}$ and into a $2$-generator finitely presented group $T_{\mathcal{Q}}$ are suggested. The constructed embeddings reflect questions mentioned by Johnson, Bridson, de la Harpe in the literature in late $1990$s about possibility of such embeddings for $\mathbb{Q}$. Technique used here is based on the methods with integer-valued sequences sets used by Higman, and with specific free constructions of groups, including free product with amalgamation, HNN-extension, an auxiliary technical structure of $*$-construction, etc.