In cold nuclear matter, the shifts in the mass and current-meson coupling as well the vector self energy of the exotic $X(3872)$ are calculated using the diquark-antidiquark current within the framework of the in-medium two-point QCD sum rule. At the rest frame of the medium, the three momentum of the considered particle is fixed to remove the contributions of the particles with negative energy. In the calculations, we include the in-medium condensates of quark-quark, gluon-gluon and quark-gluon. It is observed that, the shift due to the nuclear matter is negative and is about $25\%$ when the saturation density is used. Such shift is considerably large and comparable with the nucleon' mass shift due to the nuclear medium. The negative shift in the current-meson coupling due to nuclear matter is approximately $10\%$. At the saturation density, the vector self energy of the exotic $X(3872)$ state is found to be $Σ_\upsilon=1.31$ GeV. It is shown that the mass, current-meson coupling and vector self energy of $X(3872)$ strongly depend on the density of cold nuclear matter.

11 Pages, 3 Figures and 1 Table. A typo was corrected