In very loose terms (you can definitely ask if you'd like a more technical answer):

Model Convergence- the model is estimated in a way that uses multiple runs (think successive approximations). If a solution is available, the model will converge on a solution. If, for one reason or another, the model cannot converge, then it means that an "appropriate" solution was not found during the estimation procedure. Be cautious, though, because some programs will still give you model output with an error message for nonconvergence (the solutions aren't appropriate, so you need to figure out what's going on).

Model Fit- the degree of how "compatible" the model is with the data (maybe thought of as how well the model explains or predicts the outcome variable).

QIC is similar to the AIC for model fit-- smaller is better. The QIC, as I understand it (from a text I just referenced), is utilized to help determine the correlation structure with better fit. However, caution should be applied, as the book (Allison's Logitic Regression in SAS) indicates that the QIC can be misleading in that nonsensical correlation structures (per theory or prior knowledge) can appear to fit the data "better". Use subject matter knowledge when using the QIC to evaluate the correlation structure if there is a disagreement between structures.