The objective c'*g is linear, and the constraint c'*A*c/2 ≤ F is quadratic, so this is a quadratically-constrained problem, not linearly-constrained QP. The corresponding example in the JuMP doc would be this one.
If A is symmetric positive semi-definite, then the problem is convex and solvable fairly efficiently by several solvers such as Ipopt, CPLEX, Gurobi, Xpress, etc.
If A is not symmetric positive semi-definite, then the problem is non convex (and much more difficult to solve exactly). In that situation, Ipopt should give decent solution, though no guarantee of feasibility / optimality. Gurobi can solve non-convex QCPs to global optimality, potentially taking some time if the problem is large.