When you're the Queen of England, there is no special British Queen passport. You do not need one because all British passports are issued in her name.
Travel and Leisure cites that the official website of the British Monarchy mentions that the Queen "does not require a British passport" because the "Royal Arms," the crest of the British Royal Family, mentions that all who bear the insignia could pass freely "without let or hindrance." The British Queen passport, or likely "Her Majesty's passport," is non-existent.
Therefore, Queen Elizabeth of England does not need a passport to travel the rest of the world. She is also free not to use a driver's license should she want to drive a vehicle or even be prosecuted by law. Before you react violently to this, it would be wise to note that the Queen always acts in terms of British laws to set an example for her citizens.
According to India Today, the funny thing is everyone else in the Royal Family needs their passports to travel to different countries. Everyone else in the family needs to renew their passport from time to time, but they are given priority clearance compared to normal British citizens. The Queen of England needs not to renew her citizenship being the first citizen of Britain.
US citizens could be considered royalties traveling to the US Virgin Islands including St. Croix, St. John and St. Thomas without any passports. However, this is because the islands are within range of US national territory. Still, these islands feel foreign with the Coral World Ocean Park in St. Thomas or its Majestic Magens Bay.
But then again, these are perks and privileges for someone who is a major political character and who makes difficult decisions on an almost daily basis. As the British Queen uses her presence to political ends that require utmost priority, having no passport needed to travel speeds up her travel time.