I have a friend who had come to Qatar on a visit visa in 2006  and had to visit ( Hamad hospital) HMC emergency as she was having some pregnancy related problem. She did not have a RP or a health card.

She was admitted immediately at the hospital and ultimately had a cesarean section at HMC. However since she did not have a health card she had to pay for the hospital charges which came to around QR 13000 in total. The charges for normal delivery are ofcourse much lower.

She had come to Qatar to spend some time with her husband who had an RP in Qatar. I however do not know if u are able to avail of treatment if you are under fathers sponsorship.