If we plastically deform two compounds of different crystal structure (eg: cubic and monoclinic) to the same extent, then will there be any difference in the defect density in these two materials? If yes, then what is the role played by the crystal structure here? Does the symmetry plays any role ? (Cubic having higher symmetry than monoclinic).
In fact yes. However there are several defects. But let us concentrate on dislocations or partial dislocation. The longer the possible Burgers vector the higher the energy for a generation of this dislocation and the earlier a material fails, mainly be fracture. These materials are typically called brittle. If you look for intermetallics with quite complex structures - even cubic - these phase do not form dislocations since the energy for the formation of defects is too big. There is of course some kind of symmetry influence but this is again only a rule since it only tells you that the number of equivalent slip systems is clearly higher than for low-symmetric materials. The deformation along certain directions can be easier for lower symmetric materials (this is a question of the crystal structure) but a complex deformation is for high-symmetric phases easier. Please consider also different bondings which influence the deformation, e.g layer silicates. They are often of lower symmetry but their deformation behaviour is not that bad. Concluding, there are perhaps a few rules but not a general.
Finally some remarks to the term plastically deformed. Theoretically, after plastic deformation not even a single dislocation is still within the deformed crystal, since plastic deformation is defined by no volume change. You have a changed shape but the volume is exactly at in the beginning, Unfortunately this is often mixed up with a popular scientific understanding of deformation. This kind of practical deformation needs a change of shape and volume so that dislocations are still inside a crystal. Therefore you perhaps need to reformulate for question since in this theoretical sense the answer of your question is no since at least for dislocations one an say: it doesn't matter, which symmetry you are interested in. After plastic deformation no dislocations exist.
I agree with the above but would like to point out the difficulty of plastically deforming some non-cubic materials. Most cubic materials possess sufficient independent slip systems (5) to accommodate any arbitrary plastic strain. They can also generate the large numbers of dislocations required by easy Frank-Read source generation. When deformed they leave a significant number of "redundant" dislocations. However most non-cube materials do not have the required number of systems on which dislocations can glide to accommodate plastic strain. They therefore also use twinning systems (and sometimes local cracking) and often require higher applied stresses. The residual defect density is difficult to estimate.