For me, firing both simultaneously is a matter of tensing up the limb.
I used to have that problem a lot. Basically whenever I tried to do something at the limits of my strength, "trying really hard" meant I ended up tensing up other muscles in the same general area, including the antagonist. Obviously that was counterproductive.
It actually took me years to get out of the habit of firing antagonists in those situations. However, I think it's possible to learn it more quickly. Start by doing a movement that uses the same muscles, but requires much less strength. If possible, feel both muscles while doing it, so you can feel if the antagonist is tensing up. Get used to the feeling of using the agonist and not the antagonist.
In this case, one might start by putting your hands on the quads and th hamstrings while seated, then maybe use the quads to lift the lower leg from the floor without tensing the hamstrings. Get used to that feel, then try to do the same thing - gradually activating the quads without tensing the hamstrings - on your way into the bridge.