I read in the news recently that Boris Johnson would need the approval of 2/3 of the House of Commons if he wants to call a general election before 'Brexit Day'. I am curious as to why this is the case. The obvious answer is 'because those are the rules of Parliament'.
More specifically, section 2 of the Fixed Term Parliaments Act 2011 (FTPA) states:
(1) An early parliamentary general election is to take place if—
(a) the House of Commons passes a motion in the form set out in subsection (2), and
(b) if the motion is passed on a division, the number of members who vote in favour of the motion
is a number equal to or greater than two thirds of the number of seats in the House (including
(2) The form of motion for the purposes of subsection (1)(a) is—
“That there shall be an early parliamentary general election.”
However, what would happen if he just resigned as Prime Minister
By convention, there must always be a PM. The Queen would have to appoint another one, based on who is likely to be able to survive the confidence of the Commons.
and the entire Conservative Party simply refused to do their jobs (i.e. turn up to the commons to present/vote on legislation)? Would that not cause a crisis in Parliament that would effectively force an election anyway?
If the government didn't want to conduct any parliamentary business but still had a majority, they could vote to adjourn the Commons for a long period - or, as we've seen recently, prorogue Parliament.
If parliament is adjourned or prorogued, the government is, in theory, safe from a confidence vote.
Perhaps somewhat related: is there any requirement for a quorum to be present in the House of Commons, in order for business to proceed?
As CoedRhyfelwr mentions in their answer, the quorum for a division is 40*. However, many decisions in the Commons are done by a voice vote, for which there is no quorum. For example, adjournment debates are sometimes attended only by the Speaker (or a deputy), the MP who secured the debate, the minister responding, and a whip. At the end of the debate, the adjournment motion is then agreed by a voice vote - with 4 people present.
(* For a vote for closure (i.e. ending a debate), standing order 37 requires that there be 100 voting in favour of the closure motion.)
One of the comments below suggests that a general election can in fact be called by a simple majority via an 'ordinary law', bypassing the 2/3 majority requirement. Is that true and if so, why does the media seem to be fixated on this '2/3 majority'?
The FTPA, like any other act of Parliament, can be amended by a new act. So a government could attempt to amend this (permanently, or just for one election) by changing the 2/3 requirement. Votes on a bill just require a simple majority, but there is a risk: bills can be amended, and it's possible that the opposition could add clauses to the bill that a government without a majority might not be able to overturn - and that assumes that the government has enough votes to pass such a bill at all.