In theory, only the Queen can prorogue Parliament:
The Queen formally prorogues Parliament on the advice of the Privy Council. https://www.parliament.uk/about/how/occasions/prorogation/
Based on the answers to Can the Prime Minister prorogue Parliament?, my understanding is that, in practice, the power is held by the government and the PM's "advice" is actually mandatory and they can prorogue Parliament whenever they want.
But, what would be involved in overruling the Queen? If, the Queen refused to consent, could the PM just ignore her? Or would Parliament have to pass a law overruling her or stripping the Queen of that power?
I ask because brexiteer PM candidate Dominic Raab has proposed proroguing Parliament to prevent them from stopping a no-deal Brexit. If he believes that he can't get the votes to enable a no-deal Brexit through normal means, it seems unlikely he could get the votes to modify the Queen's power to accomplish the same thing.