We have paired samples where cell lines from three individuals were treated with a peptide. Treatment with peptide compared to controls using DESeq2 default Wald's test did not bring any significant DEGs (genes padj > 0.9).

model: ~ individual + treatment

So I tried to test differential expression with LRT

full model: ~ individual + treatment
reduced model: ~ treatment

and got significant DEGs showing consistency with the literature.

Is it correct to use LRT here? I would really appreciate your comments.

The above LRT is not what you want. It is testing whether the individuals have different baseline, controlling for treatment effects. You want to exclude the variable of interest in the reduced design.