Yes, but constitutionally, she can only exercise influence over her Prime Minister and her government in private. She cannot do so in public.
The formula used to describe the situation is that the Queen has the right to be informed, to be consulted, to encourage, and to warn the Prime Minister in private on all laws. They have a weekly meeting expressly for this purpose which is completely private. During that meeting she is allowed to - in fact is supposed to - counsel the Prime Minister and tell him if she believes a law is a bad idea, and he or she is required to listen to her and consider what she has said. But the Prime Minister makes the decisions.
Prime Ministers have modified their policies in the past, apparently based on such meetings. But if they choose not to, that is their right. And the Queen cannot gainsay them in public. They are Her Majesty's Government - constitutionally a situation would be intolerable if the Queen campaigned against government policy. And constitutionally, ultimate power over legislation rests with Parliament not the Queen. Except on a very small number of constitutional matters. The Queen does have real power - but only in cases where Parliament can't do something (for example, if Parliament can't clearly choose a Prime Minister, the Queen would do so - and has used this power, twice).
Surely, as Defender of the Faith, she should have fought this vigorously; unless she truly does believe the State is, and should remain more influential in the lives of the nation than the natural law of the Supreme Being. It is one thing for America to grapple with the church/state question; it is another for a nation with an established Christian church to throw it all in the Thames.
To all those la-di-dah dismissers of the negative consequences of this decision, note that it has taken 500 years since the English church was formed to suit a lecherous King Henry XIII until this decision, which puts not only the final nail into freedom in Christ; it also drives a stake through the heart of the corpse. In this "brave" new future, children will be nothing more than prey; and women will return even farther into chattel status than they were before they started pressing for "liberation" from their traditional roles. The only difference being that before, they were chattel who were responsible for modesty and chastity. Now, they will be chattel who are expected to dress and act like whores and stand aside without protest the minute their baby daddy wants a different concubine or a boyfriend. Sad, sad day for the ghosts of Western Civ.