Two-way ANOVA with count data Can we report a Two-way ANOVA with count data? If Yes, What are your references?
If No, Why?
For example:
Factor A in 4 level and Factor B in 3 level and our responses are number of patients.  
 A: IF your count variable can be modelled as a Poisson distribution, you can use a GLM---Poisson regression, or some variants thereof. If the distribution only is "similar to" a Poisson, you can use quasi-Poisson regression or negative binomial regression.
OR, in some cases (especially if your only covariable is group membership) you can use ANOVA, after using a variance-stabilizing transformation. For the Poisson, that is the square root.  In most cases today, it will be better to use a generalized linear model (Poisson or negbin) than transforming. For some strong opinion see here.
A: A little bit different answer from Glen_b's (though I agree with his).
Count data is usually best modeled with distributions like a binomial (where there is a maximum count), Poisson, or negative binomial (or others).  In these distributions there is a relationship between the mean and the variance.  ANOVA models assume normal distributions and equal variance, both of which are violated if the "truth" is one of these other distributions.  That being said, the binomial, Poisson, and negative binomial can all be approximated by a normal distribution under certain conditions (large sample sizes and means far from the boundaries), so if your counts are high enough (and the resulting variances are not too different) then an ANOVA model may be a reasonable approximation (but the other models will still be better).
A: 
Can we report a Two-way ANOVA with count data? 

Of course you can, the bigger question is whether you should.
In general, you should not - but instead you should do a similar analysis suitable for the particular kind of count data you might have.

If No, Why? 

Because they don't satisfy the assumptions of anova. You have heteroskedasticity (variance is related to mean), skewness (counts can't be negative, and in small average counts the skewness will be substantial), and discreteness (again, with small average counts the impact can be substantial).

our responses are number of patients

"number of patients" doing what? Is this number of patients having some characteristic out of some total (like number out of the total exposed who responded to treatment or showing some symptom, say) or something else?
You may need some binomial or Poisson GLM (or perhaps negative binomial or some other model would be more suitable), or it may be you can set it up as a chi-square test.
That said, sometimes an ANOVA can work fairly well. If the counts are large enough that the above issues with skewness/closeness to 0 don't impact much, and you're mostly just interested in whether you can reject the null, the test should have close to the desired significance level under the null (heteroskedasiticity should only crop up under the alternative). 
