I am trying to calculate QAIC of DLNM models with code offered in (https://github.com/gasparrini/2013_gasparrini_BMCmrm_Rcodedata/blob/master/01.prep.R).

but I am confused about the difference between this and the formula of QAIC from other R package as in the answer to this question (Akaike criterion (AIC) for quasi-Poisson models).

The former adds dispersion parameter to the likelihood term of AIC formula while the other to the df term, although increasing dispersion would lead to decrease of QAIC in both.