Answer:
I would not use either of those constructions. The verb 'comply' is followed by 'with', so the noun 'compliance' should be as well.
'Compliance with this new rule is expected from next Monday.'
'The company demonstrated their compliance with the new legislation by ...'
(But better would be 'The company showed how they complied with the new legislation by ... ', or a similar construction.)
I suppose you could say 'The compliance of the company was demonstrated by ...'; it is grammatically correct but hardly idiomatic. And it means something different from 'compliance with', of course.