The quadrature is a process – any process – of turning something into a "square". "Quadro" in Latin is "make square", "quadrus" is a "square". It comes from "quattors", four, because that's the number of vertices of a square.
So integration of a function is also known as "quadrature" because we are calculating the area i.e. looking for a well-known area (square) whose area is the same. We are turning the area into a square; so what we are doing is "quadrature". Alternatively, the numerical integration based on the Riemann integral is really dividing the area into thin rectangles and rectangles are still good enough for the word "quadrature".
In recreational mathematics, "quadrature" without adjectives (or "quadrature of a disk" or "squaring the circle") is the task invented by ancient geometers of constructing a square with the same area as the given disk. It may be proven – and has been proven since the late 19th century – that this task cannot be exactly solved with the usual geometric tools.
Similarly, the operators $x$ and $p$ are called quadrature operators (for the harmonic oscillator or any generalization) because they allow us to write the Hamiltonian $H$ as a quadratic function (of these quadrature operators). We are turning the Hamiltonian into a square $a^\dagger a$ (or a sum of squares $x^2+p^2$), so we are doing quadrature again. That's where the name comes from.
Just to be sure, functions like $y=x^2$ are called "quadratic" which is also linked to the square because they "are" the square – or, more carefully, they are formulae to calculate the area of a square.
In some disciplines, like quantum optics, this name is more popular than others but every scientist should be able to understand the word.