The question could be more precise because there is no mutual couplings between d and q axis. The equations for flux linkages in d and q axis starts from their form in the stand still frame. After transformation from this to the rotor frame (this must be done for all quantities: inductances, resistances, voltages, currents and flux linkages) the final equations for flux linkages shall be derived. Also, the systematic approach can be found in: P. C. Krause, Analysis of Electric Machinery!