Things would get a bit interesting, and changed in 2011. Precedent is now of questionable validity.
Before 2011, the government needed to be able to win any confidence vote in the House of Commons. This could be an explicit vote on confidence in the government, but it could also be any vote the government designates as a confidence vote (all supply bills were in the latter category). That means that the government could say "if Parliament doesn't pass this, we're resigning." Losing a confidence vote normally meant elections were called; the Queen could invite someone else to form a government instead of calling elections, but that was generally not done. The government could also just go ahead and ask for new elections without bothering with a confidence vote; again, that could be denied, but likely wouldn't be.
What governments could do as an alternative to new elections was get a confidence-and-supply agreement with another party, under which the government would make policy concessions to that party and that party would vote with the government on all confidence votes. Since substantive votes could be confidence votes, that means that the government could use confidence-and-supply agreements to get votes on actual issues they cared about.
Much of the above is irrelevant now. The Fixed-term Parliaments Act 2011 says that Parliaments last five years; they cannot be dissolved early, with two exceptions. First, two-thirds of the House of Commons can vote to dissolve. Second, an explicit motion of no confidence can lead to dissolution. In the latter case, it must be explicit; there are no more confidence votes on actual issues, which means the government can't necessarily use a confidence-and-supply agreement to get its way on important policy. Also, there are 14 days after a vote of no confidence for Parliament to pass a motion of confidence, cancelling dissolution. That seems designed to give the opposition a chance to form a government without new elections being called. However, as things stand, the conventions around this are unclear. It appears that minority governments are more stable than before, but they may have a much harder time getting core policies passed. A party can't get new elections to shore up its majority.