Is it permissible for me to obtain subcontracting work in government projects that I supervise, knowing that there is no mediation or payment of money involved, and a price quote is submitted just like any other establishment?

It is not permissible for a project supervisor to accept and complete the work of contractors, as this creates a conflict between the duty of supervision and the execution of the work, and out of concern for favoritism and overlooking issues. The employer must be informed of this; if they grant permission, there is no harm; otherwise, it is not allowed.