We introduce a $q$-deformation that generalises in a single framework previous works on classical and enriched $P$-partitions. In particular, we build a new family of power series with a parameter $q$ that interpolates between Gessel's fundamental ($q=0$) and Stembridge's peak quasisymmetric functions ($q=1$) and show that it is a basis of $\QSym$ when $q\notin\{-1,1\}$. Furthermore we build their corresponding monomial bases parametrised with $q$ that cover our previous work on enriched monomials and the essential quasisymmetric functions of Hoffman.