ACF-PACF could be the diagnostic model of seasonality? If seasonality is not shown in the acf-pacf plot, can I conclude that the seasonality does not exist? Or Can I say that the seasonality exists based on the monthly incidence data although there is no seasonality in the acf-pacf plot?
 A: Did you plot the monthly incidence data versus month of year using side-by-side boxplots (presuming you have multiple years worth of monthly incidence data)? This would allow you to assess the presence of seasonality in a more intuitive fashion. See this link for details: itl.nist.gov/div898/handbook/pmc/section4/pmc443.htm. 
This link is also useful, as it gives examples of what additive or multiplicative seasonality would look like if you examined a time series plot of your data: anomaly.io/seasonal-trend-decomposition-in-r. See the plots in Step 6 of the link for a nice illustration of the two types of seasonality. 
If you detect seasonality in the side-by-side boxplots or in the time series plot of your data, then the acf-pacf plot will reflect that seasonality as well. 
If you don't detect seasonality in the side-by-side boxplots or in the time series plot of your data, then you won't detect seasonality in the acf-pacf plot either. 
You can't say seasonality is present just because you have monthly data - you need to have clear evidece from your plots that seasonality is present before you can make your claim. It is possible for monthly data to not exhibit any seasonality. It is also possible for monthly data to exhibit seasonality. Examining all plots carefully will reveal which situation you are dealing with.
A: I think you can go as far as saying that you didn't find evidence of seasonality after inspecting the ACF and PACF plots. Making the claim that seasonality does not exist is going another step beyond that.
You can't claim that seasonality exists just because you have monthly data. You can hypothesize that there is seasonality in the monthly data and have good reason for doing so, e.g. if you are dealing with ice cream sales. However, there are cases where this type of assumption wouldn't be reasonable, e.g. if you are dealing with monthly exchange rates.
A: Simple time series identification tools that assume that seasonality is auto-projective are often simply overmatched by the data. Check the assumptions underlying all tools.
Many time series are affected by monthly determinstic structure e.g. a Dec effect or a June effect .
In both cases the original acf/pacf seasonality suggestion/hint may be obfuscated by unusual values (pulses) or level/step shifts or local time trends or changes in model error variance or parameters over time.
Build your model iteratively following this paradigm https://autobox.com/pdfs/ARIMA%20FLOW%20CHART.pdf ultimately yielding a white-noise error process.
