Can the UK parliament push a legislation, actively opposed by the Queen? Can they overcome the Queen's veto?
This question is tricky.
From what I can tell, there are 3 possible interpretations/meanings, depending on how much you like technicalities:
NoThe Queen is not separate from the Parliament; in fact, she is its head. It is only when the Queen has granted a bill her royal assent that it becomes an Act of Parliament. So you could say that, if the Queen does not agree, then the Parliament is not actually pushing any law.
YesIf you ignore that technicality and go with the standard view of defining Parliament as just the House of Commons & the House of Lords (and in later times, principally the House of Commons alone), then the Queen retains the theoretical power of refusing to give royal assent to a bill approved by both Houses or the House of Commons alone.
Theoretically yes, but at a considerable costIf you go to the totally practical level, you see that the last time royal assent was denied was in 11 March of 1708. (Though in 1914, the King sought legal advice about whether he could deny assent.) At this point in time, trying to deny royal assent would amount to a very extraordinary intervention by the Monarch, which could have severe political consequences; while not really being impossible, it could be considered on par with a coup d'etat from the Monarch and could lead to severe political repercussions.