When travelling overseas, The Queen does not require a British passport. The cover of a British passport features the Royal Arms, and the first page contains another representation of the Arms, together with the following wording:
'Her Britannic Majesty's Secretary of State requests and requires in the name of Her Majesty all those whom it may concern to allow the bearer to pass freely without let or hindrance and to afford the bearer such assistance and protection as may be necessary.'
As a British passport is issued in the name of Her Majesty, it is unnecessary for The Queen to possess one. All other members of the Royal Family, including The Duke of Edinburgh and The Prince of Wales, have passports.
In realms (Commonwealth countries where The Queen is Sovereign), a similar formula is used, except that the request to all whom it may concern is made in the name of the realm's Governor-General, as The Queen's representative in that realm. In Canada, the request is made in the name of Her Majesty by the Minister of Foreign Affairs.