Origins of the name "Q" and "R" for cofibrant and fibrant replacement functors. In a model category $\mathscr M$ (in the modern sense, i.e. closed and with functorial factorizations), there is a notion of fibrant and cofibrant replacement functors.
Specifically, for any object $x$ of $\mathscr M$, the $(\mathrm{Cofib}\cap W, \mathrm{Fib})$-factorization
$$ x \to x' \to 1  \qquad \text{($1$ final object)}$$
gives rise to a fibrant replacement functor $R \colon \mathscr M \to \mathscr M$. Dually, the $(\mathrm{Cofib}, \mathrm{Fib}\cap W)$-factorization
$$ 0 \to x'' \to x \qquad \text{($0$ initial object)} $$
gives rise to a cofibrant replacement functor $Q \colon \mathscr M \to \mathscr M$.
I did not choose the letter $Q$ and $R$ randomly. They are all over the literature (Hovey, Goerss-Jardine, etc.). Why those letters? They hardly seem natural...
 A: As Quillen passed away a few years ago we may never get a definitive answer. However, some thoughts (some collected from the comments):


*

*Q is the letter just before R in the alphabet, and the cofibrant/fibrant replacements fit in a diagram like $Q_X \to X \to R_X$, in the same order as in the alphabet (see user314's comment). More generally one puts cofibrant objects in the left slot of the hom functor and fibrant object in the right slot. This remark can probably be used to explain the choice of the other letter as soon as one letter is chosen.

*The letter R might stand for (fibrant) Replacement.

*The letter Q might stand for Quasi-free. A quasi-free Lie algebra (for example) is a dg-Lie algebra which is free as a graded Lie algebra, equipped with an additional differential. Under some conditions these are cofibrant in the model category of Lie algebras. Quillen is one of the founders of rational homotopy theory, where such Lie algebras play a crucial role. (However his paper Rational Homotopy Theory was published two years after Homotopical Algebra, and I don't know enough about the history to know if he already was working with Lie algebras in this way in 1967.)

*The word cofibration starts with a "Q" sound.

*The letter Q might stand for Quasi-functor. Quillen notes in the proof of Proposition 5 in Chapter II, Section 2 that $X \mapsto Q(X)$ and $X \mapsto R(X)$ define "quasi-functors".

*The letter R might stand for Reflection, see Zhen Lin's comment.


Or it's possible that the letters were chosen at random and the curtains were just blue. I don't know.
