Based on our experience, The excitation wavelength for ZnO should be  270 nm for thin films photolumeniscence studies.

get uv vis spectra of your sample.... and use the peak value of the absorption (wavelength) as excitation wavelength for getting fluorescence .......... it should be around 300nm

Before taking luminescent spectra, it is advisable to take absorption spectra of ZnO thin film. The absorption peak wavelength should be used as excitation wavelength for photo-luminescence  spectra.

For ZnO the excitation wavelength of the light should be less than its band gap, for example in south Korea most of the labs will use 325 nm wavelength laser for excitation and some time we can also use 360 nm.

the wavelengths suggested by the other responders sound ok.  In dealing with a thin film you might have to do the measurements with 45 degree incidence angle in order to maximize the signal.  And  secondly you might find it necessary to do a front surface detection (excite and detect from the same direction) in order to minimize the effects of absorption - ZnO is a good absorber in the UV, it has an optical gap around 405nm which means it absorbs strongly at shorter wavelengths.  It does fluoresce.

The Excitation wave length for ZnO should be 375 nm

Do the UV-VIs measurement of your sample. You will find the band edge absorption peak. Now you can select the excitation wavelength slightly higher than the absorption peak for fluorescence measurement.