Not currently: https://github.com/jump-dev/MathOptInterface.jl/issues/1390.
tl;dr: there are a bunch of non-standard extensions to MPS to support quadratic. We just haven’t implemented them yet.
You can always use the solver’s native writer:
model = direct_model(Gurobi.Optimizer)
grb = backend(model)
GRBwrite(model, "model.mps")