The existence of permanent electric dipole moments (EDMs) and magnetic quadrupole moments (MQMs) violate both time reversal invariance (T ) and parity (P ). Following the CP T theorem they also violate combined CP symmetry. Nuclear EDMs are completely screened in atoms and molecules while interaction between electrons and MQMs creates atomic and molecular EDMs which can be measured and used to test CP-violation theories. Nuclear MQMs are produced by the nucleon-nucleon T, P -odd interaction and by nucleon EDMs. In this work we study the effect of enhancement of the nuclear MQMs due to the nuclear quadrupole deformation. Using the Nilsson model we calculate the nuclear MQMs for deformed nuclei of experimental interest and the resultant MQM energy shift in diatomic molecules of experimental interest 173 YbF , 177,179 HfF + , 181 TaN, 181 TaO + , 229 ThO and 229 ThF + .The observed matter-antimatter asymmetry in the universe is an important open question in modern physics. Three necessary conditions were postulated by Sakarhov[1] including the requirement that combined charge and parity (CP ) symmetry is violated. While the current standard model (SM) includes a CP -violating mechanism through a CP -violating phase in the CKM matrix [2] this alone is insufficient to account for the observed matter anti-matter asymmetry by several orders of magnitude (see e.g. Refs. [1,[3][4][5][6][7]). Therefore, other sources and mechanisms of CP -violation beyond the current SM must exist and investigating these will give insight into new physics.The violation of CP symmetry was first detected in the decay modes of the kaon system [8] and more recently in the B meson sector [9, 10] however detection of CP -violation in other systems has not been confirmed. By the CPT theorem a mechanism which violates combined CP symmetry must also violate time-reversal (T ) symmetry. Therefore, the existence of permanent electromagnetic moments which violate T symmetry is a promising avenue for constraining theories which incorporate a higher degree of CP -violation than the SM such as supersymmetric theories which has already been tightly constrained by current experimental limits for electric dipole moments (EDMs) [5,11,12].CP -violating permanent electrodynamic moments are expected to be observed in composite particles and systems such as atoms, nuclei and baryons and interpreted as parameters of CP -violating interactions in the lepton and quark-gluon sectors. In this paper we focus on the magnetic quadrupole moment (MQM) of the nucleus in particular, which is the lowest order magnetic moment that is forbidden in quantum systems by the time reversal invariance (T ) and parity (P ). For an in-depth review on symmetry violating electromagnetic moments including the MQM see Ref. [5,[13][14][15][16][17]. The MQM of composite systems such as the deuteron [18] have previously been investigated. The search for MQM in comparison with the electrostatic T, P -violating moments (EDM, Schiff and octupole moments) may have the following advantages:• The n...