I tried some Options, but with no result. Seems indeed to be a bug. A very brute and simple method yields a reasonable result:

qq1[X_, n_] := Sum[X/n QQ[j X/n], {j, 0, n}]
Plot[qq1[x, 50 x], {x, 0, 10}]

Interesting: this works as well

f[x_] := If[Sin[x] < 0, 0, Sin[x]]
Plot[NIntegrate[f[u], {u, 0, uu}], {uu, 0, 20}]

Hold doesn't work for me. The plot has no content, just the axes. This is a really strange problem.

I would like to greet all of you. Your comments and suggestions are very valuable, because my "mother milk" was FORTRAN IV on IBM7040. Now, after retirement, I have Mathematica Home Edition and this is a different kettle of fish.

My experiments:

NIntegrate[Hold[If[EvenQ[Floor[t]], 0, 1]], {t, 0, A}]          works

NIntegrate[If[EvenQ[Floor[t]], 0, 1], {t, 0, A}]                    does not work

NIntegrate[Hold[If[Mod[Floor[t], 2] == 0, 0, 1]], {t, 0, A}]   works

NIntegrate[If[Mod[Floor[t], 2] == 0, 0, 1], {t, 0, A}]             works too

It seems to me that problem is connected with function EvenQ as can be deduced from yours experiments too..