In this work, the problem of event-triggered mechanism (ETM) and dissipative controller design for discrete-time (DT) two-dimensional (2-D) switched Fornasini-Marchesini (FM) systems is addressed. First of all, the definitions of 2-D (H,L,M,∊)-dissipativity and exponential stability for the considered systems are introduced. Then, the ETM is constructed to determine whether the signal is transmitted or not for the objective of control. Meanwhile, the definition of event-triggered sequence is given based on the line of cross-cut technology. Subsequently, the event-triggered dissipativity analysis and dissipativity-based control problems are solved. And corresponding sufficient conditions are obtained, under which the closed-loop 2-D switched system is exponentially stable and strict dissipative under the ETM. Finally, an illustrative simulation example is employed to verify the availability of the proposed controller design result, and the influence on the dissipativity performance index (DPI) is explored while the triggered parameter is changed.