The situation you're asking about is a Hung Parliament and is fairly common. While the Queen can't simply take over government if parliament is hung, she (or rather her ministers) can use Orders in Council to handle any emergencies should they arise. Her ability to intervene in the parliamentary deadlock itself is more complex, and there's an illustrative case where this happened not in the UK but in Australia known as The Dismissal.
Let's look at both possibilities.
Order in Council
@WS2 pointed this power out in a comment.
Should Parliament be in turmoil and an emergency arise, the Queen or one of her Ministers, which includes the Prime Minister, can use her Royal Prerogative (see below) to issue an Order in Council. This is law by royal decree bypassing the usual legislative process. However it is still subject to judicial review, and the legislature can later pass law overriding the degree.
The Civil Contingencies Act 2004 formalized the process of when an Order in Council can be used for an emergency, how long the order would last, and what constitutes an emergency.
- threatens serious damage to human welfare in the UK
- threatens serious damage to the environment of the UK
- war, or terrorism, which threatens serious damage to the security of the UK
A hung parliament itself is not an emergency. But should an emergency arise while parliament is hung, then the country could be governed by Orders in Council. For example, if the UK were to crash out of the EU and if this caused a financial panic this could qualify as an emergency as a "disruption of a supply of money" (19.2.e). If parliament was hung, an Order of Council could be allowed because "the existing legislation cannot be relied upon without the risk of serious delay" (21.5.a). Such an Order would lapse after 30 days (26.1.a).
Now a bit about what a hung parliament is, how governments are formed, what the Queen's powers are, and the example from Australia known as The Dismissal.
The situation you're asking about is a Hung Parliament. They're not uncommon and usually resolve themselves within a few days. One occurred in 2017 when May's gamble to increase their majority failed and her Conservative Party lost their majority, but retained a plurality. Negotiations occurred between the Conservatives and the small DUP to try and form a majority.
Based on her confidence in these negotiations, May formed a minority government. This is governance not only by a party which does not hold a majority, but also where their governing coalition is not particularly strong. The DUP will support the Conservative government on votes of confidence and budget votes critical to hold their government together, known as confidence and supply, but reserve the right to vote as they please on other matters.
There's technically no need to form a governing coalition, members are allowed to vote as they please, but without one its unlikely the Prime Minister would survive a vote of no confidence. Let's talk about that.
Vote Of No Confidence and the Royal Prerogative
Technically, the Queen chooses their Prime Minister for their government. This is all part of the Royal Prerogative; the idea that all governmental power emanates from the sovereign who gets it from God. It is Her Majesty's Government.
In reality the Royal Prerogative is a mish-mash of formal rules and traditions. In practice these have all been delegated to the "advice" and "request" of her ministers and things would have to go very pear-shaped indeed for the Queen to go against her ministers. The Queen could technically choose anyone they wanted as Prime Minister. In practice the sovereign takes the advice of the outgoing Prime Minister who to appoint next.
It's called a "vote of no confidence" because technically it demonstrates the House of Commons "has no confidence in Her Majesty's Government", that is the government formed by the Prime Minster who is technically chosen and empowered by the Queen. This represents a check on the power of the sovereign by the people to the extent that the Queen cannot step foot inside the House of Commons.
Upon a vote of no confidence, the Fixed-term Parliaments Act 2011 requires that, unless a new government (whether of the same party as before or not) wins a vote of confidence within 14 days, a general election must be held.
In practice, the Sovereign interfering with parliament would cause turmoil and a Constitutional Crisis. It would throw the country into one of those grey areas that isn't well defined.
Finally we get to Australia and The Dismissal.
In Australia, Canada, Wales, Scotland, and Northern Ireland the Queen retains significantly more power over their parliaments. Usually the Queen's power is delegated to a Governor-General. And, as in the UK parliament, usually the Governor-General's role is a formality and acts at the "request and advice" of the Prime Minister.
In 1975 Australia found itself with a slim, corrupt Labor majority controlling the House of Representatives, but the opposition controlling the Senate. The Senate blocked important budget bills needed to fund the government until new elections were held, not unlike the current deadlock in the US.
The Prime Minister met with the Governor-General advising the Governor-General to overcome the deadlock with an election of half the Senate. Instead, the Governor-General dismissed the Prime Minister eventually leading to a Double Dissolution of both the House and Senate.
While the Prime Minister was being dismissed, the Senate was resolving to end the deadlock. Oops.
The Australians were, understandably, quite miffed at the interference in their government by the Queen's representative. The Governor-General of Australia was seen as a courtesy and formality. He had intervened against the advise of the Prime Minister, and far, far too soon. Had he waited just an hour the crisis would have resolved itself.
While the Governor-General's actions were embarrassing, there was no major change. There were some protests and a few minor changes to Australia's Constitution, but the situation remains the same. An attempt in 1999 to formally turn Australia into a republic independent of the Queen failed 45% to 55% against.
There are some other, now almost entirely formal, ways the Queen can intervene in parliament.
The Royal Assent
The Royal Assent is required on all legislation enacted by parliament before it becomes law. Prior to 1541 it was required that the sovereign personally read publicly the whole act. King Henry VIII decided that publicly reading out his own order to execute his fifth wife for not telling him about her previous lover wasn't such a hot idea, so the Royal Assent by Commission Act 1541 granted this power to specially appointed commissioners. The Royal Assent Act 1967 changed it so parliament is simply notified of the assent.
The last time the Royal Assent was withheld was in 1708.
Prior to 2011, the sovereign technically retained the sole power to dissolve Parliament. The Prime Minister would request that the sovereign dissolve Parliament and hold new elections, but that became increasingly a formality. The sovereign could, on certain agreed upon conditions such as time of war, refuse the request.
The sovereign's ability to dissolve the UK Parliament was formally removed in the Fixed-term Parliaments Act 2011.