You have 2 degrees of freedom for your variable A, because (you should know) it is a categorical variable with 2 levels. So if you report the F value for this, it is basically the variance explained for all the levels under A. It depends on what A means, for example, if A is a category "region" with two possible values, then it is normally to report the F value for "region" with 2 degrees of freedom.
Some made-up data, where A is categorical with 2 levels, it gives you npar=2.
df = data.frame(A=factor(sample(0:2,100,replace=TRUE)),
df$y[df$rand==1] = df$y[df$rand==1]+3
fit = glmer.nb(y ~ A+B+C + (1|rand),data=df)
Analysis of Variance Table
npar Sum Sq Mean Sq F value
A 2 2.82093 1.41047 1.4105
B 1 0.05069 0.05069 0.0507
C 1 0.87359 0.87359 0.8736
Next, since you fitted a linear mixed model, so the anova function doesn't provide a p-value (for a good reason) because it is not clear in a model with random effects, whether this provides a reliable p-value, quoting from FAQ section for glmm :
it is not in general clear that the null distribution of the computed
ratio of sums of squares is really an F distribution, for any choice
of denominator degrees of freedom.
You can also check this post or the help page for other possible alternatives, but I think you need to be careful about doing this. If you really need a p-value for some purpose, it might be better to revert to a fixed effect only model.
In your case, this post would be relevant as well, and if you provide more context on what you are testing, maybe other users can also suggest a more appropriate way of reporting your results