Is it permissible to use my friend's health insurance card for a medical examination due to difficult financial circumstances and inability to conceive, knowing that this is without her knowledge?

It is not permissible to resort to deception and fraud in people's wealth, and that includes using your friend's insurance card if the terms prevent it. One must fear Allah and seek His help in seeking healing, for He alone is the Healer. Allah Almighty says: "And when I am ill, it is He who cures me," and "And whoever fears Allah - He will make for him a way out And will provide for him from where he does not expect."