Does Gödel's first incompleteness theorem apply to quantifier-free arithmetics?
What you are referring to is called Baby Arithmetic ($\mathsf{BA}$) in Peter Smith's book An introduction to Gödel's Theorems.


  $\mathsf{BA}$'s language [has] one single individual constant $0$, the one-place function symbol $S$, and the two-place function symbols $+$ and $\times$.  [...] it lacks quantifiers and variables.


$\mathsf{BA}$ is complete, negation complete, and decidable.