Must I pay the full extra amount for a private room after a prior agreement on a reduced price for a private room at the same price as a quad room?

The contract is the law between the contracting parties, and based on this, what was initially agreed upon is what matters. If the contract stipulates your stay with your family in one room at the mentioned price, then it is not permissible to violate it, nor to obligate you to pay more than the agreed-upon price. However, if the contract does not stipulate that, and the presence of your family with you harms the other party, and you committed to him to pay the price difference, then you are obliged to bear it. If there is a dispute regarding the matter, it must be presented to the competent authorities, or you should ask people of knowledge for further clarification.