First, MIQP problems are difficult to solve. Especially ones with 354,000 binary variables! The answer you’re looking for is highly problem dependent, so if you can post a reproducible example, people may have suggestions for modeling improvements. Trying to tune the solver is probably of little benefit.
Are there any other solvers that I should try and see if they’re faster? I’m not paying for a solver, though.
You don’t have many options, unfortunately. If you’re an academic, I’d suggest Gurobi. Otherwise it’s mainly SCIP. Other options like Bonmin are likely to also struggle.
As far as I understand all quadratic programs are convex
This is not true. x^2 <= 1 is convex, x^2 >= 1 is non-convex. (The feasible region is x <= -1 and x >= 1. The range x in (-1, 1) is infeasible.) Similarly, min x^2 is convex, max x^2 is non-convex. Also, terms like x * y are non-convex.