Is it permissible to pay 10,000 Saudi riyals to a public services office in exchange for transferring me from my military post to another city?

If the transfer is rightfully yours according to labor regulations, there is no harm in obtaining it through intercession or any other legitimate means, provided no harm is inflicted upon others. Paying money to secure this right is not considered a bribe on your part. However, if it is not rightfully yours, then it is not permissible to seek to obtain it.