With the log-likelihood chi-square statistics I can compare two linear mixed models (Maximum Likelihood) and see which one is the better one. But the GEE gives Quasi Likelihood under Independence Model Criterion (QIC) and I don't see the degrees of freedom so I am unsure how to statistically test two models against each other and select the one with the best fit.
With GEE, you should do Wald tests on the coefficients. The likelihood-ratio type tests are sensitive to their assumptions such as normality of the response and the random effects. GEEs trade away the MLE efficiency for robustness (against possible misspecifications of the within-cluster correlations), and if the mixed model MLE and the GEE contradict one another, I would trust the GEE, not the MLE.
If you never came across the Wald test before, read the great exposition by Buse (1982) about the relation between different asymptotic tests.