tl;dr
Dedekind was the first to use a letter (R) for sets of rational numbers in 1872, then, starting from 1895, Peano began to use the letter r (lowercase) to denote the same set (and, from 1889, R for the set of positive rationals). Other authors proposed different letters, and only in the early forties Bourbaki introduced the letter Q (not the blackboard bold $\mathbb{Q}$). Possibly, the blackboard bold version was introduced (maybe) in the late fifties or (more likely) in the early sixties, but this is another question.
Now, some references.
Dedekind used the letter R (uppercase) for the set of rational numbers in
Stetigkeit und irrationale Zahlen (1872), $\S 3$, page 16 ("die Gerade L ist unendlich viel reicher an Punkt-Individuen, als das Gebiet R der rationalen Zahlen an Zahl-Individuen", i.e. "the straight line L is infinitely richer in point-individuals than the domain R of rational numbers in number-individuals", here the English translation).
About Peano, Wikipedia "clearly" refers to somewhere in the Formulaire de mathématiques/Formulaire Mathématique/Formulario Mathematico, where Peano actually used extensively letters to denote sets (classe) of numbers. There's no doubt about this, as this is the only work of Peano concerning the subject for the year 1895 (see here for the complete bibliography of Peano, the works for the year 1895 start at page 45). The problem is that (see the wiki page)
the five editions of the Formulario [are not] editions in the usual sense of the word. Each is essentially a new elaboration, although much material is repeated. Moreover, the title and language varied: the first three, titled Formulaire de Mathématiques, and the fourth, titled, Formulaire Mathématique, were written in French, while Latino sine flexione, Peano's own invention, was used for the fifth edition, titled Formulario Mathematico. ... Ugo Cassina lists no less than twenty separately published items as being parts of the 'complete' Formulario!
and moreover the Formulario was written by many Peano's collaborators, such as Giovanni Vailati, Mario Pieri, Alessandro Padoa, Giovanni Vacca, Vincenzo Vivanti, Gino Fano and Cesare Burali-Forti, so when one writes "in the Formulario Peano says..." one have to understand that actually one is referring to Peano or to one of his collaborators.
The following quotes are taken from the Formulario Mathematico edited in 1908, for the reason that it is the clearer and more complete exposition of the subject. One can read the 1908 Formulario here, while different editions of the Formulaire de mathématiques are on Gallica, for example here.
First Peano writes (in I.$\S 1$, of course) the symbol $N_0$, together with $0$ and $+$ as "idea primitivo" [sic], i.e. undefined primitive ideas used to define all the other symbols of the "Arithmetica", and whose sense is determined by a system of proposition, the first is:
$\cdot 0\quad N_0\ \varepsilon\ \text{Cls}$
that he read "$N_0$ is a class", and so on. In II.$\S 5$ we find $N_1=N_0+1$, so $N_1$ is the set of strictly positive natural numbers, and in II.$\S 6$ he uses $+N_0$ and $-N_0$ for positive and negative numbers, and $n$ for the union (so $n$ stands for $\mathbb{Z}$). Then in III.$\S 8$ we read
$\cdot 2 \quad R=N_1/N_1$
and
$R=$ "Numero rationale". Illo es omni expressione de forma $b/a$, ubi $a$ et $b$ es numero naturale...
i.e., $R$ denotes a rational number, and it is any expression of the for $a/b$ where $a$ and $b$ are natural numbers; clearly Peano also introduces $r=+R\cup -R\cup \iota 0$ (III.$\S 9 1\cdot 0$). Moreover he adds
$\cdot 01\quad r=n/N_1$
$r=$ numero rationale relativo
and so $r$ stands for $\mathbb{Q}$.
Finally, in III.$\S 12$ we find
$5\cdot 0\quad Q=l'`[\text{Cls}` R \cap u\ni(\exists u.\exists R=\eta u)])$
that Peano reads
$Q$, lege "quantitate reale positivo" es omni limite supero de aliquo classe $u$ de rationale, existente, et tale que existe aliquo rationale maiore de omni $u$
i.e., $Q$, that reads as "real positive numbers", is any supremum of some existing set (classe) $u$ of rationals, such that there exists some ratonal number greater then all elements in $u$; clearly in III.$\S 13$ Peano defines $q$ as $Q\cup -Q\cup \iota 0$, ans so $q$ stands for our $\mathbb{R}$.
In the Formulaire de mathématiques one can read more or less the same things (in French, of course), in particular in the index at page 56 we find "$r$ nombre rationnel".
Anyway, Peano used letters for sets of numbers before 1895, see Arithmetices principia, nova methodo exposita (1889), the Signorum Tabula at page 13 ("$Q$ quantitas, sive numerus realis positivus", i.e. "$Q$ [stands for] quantity, that is, a real positive number", while $R$ is used for positive rationals, no symbol for the rationals, positive or negative).
Summing up: yes, Peano uses letters to denote classes of numbers, but no, his use is different (indeed, antithetical) from the modern one; moreover, capital letters are used for set of only positive or only negative numbers of some kinds, while lowercase letters for either positive or negatives numbers ($n$ for $\mathbb{Z}$, $r$ for $\mathbb{Q}$, $q$ for $\mathbb{R}$ [sic]).
Furthermore, I cannot find any source independent of Wikipedia about the use of letter $Q$ from the Italian "quoziente", and moreover, in Italian the elements of $Q$ are called "frazioni" (fractions) or "numeri razionali" (rational numbers, from the Latin word "ratio"), while "quoziente" is the result of an operation. The choice of letters made by Peano is transparent: $n$ is for "numerus" (whole number, "numero" in Italian), "r" for "(numero) rationale" (rational number, but also "rapporto" in Italian, ratio in English), "q" for "quantitas" (Italian "quantità", English quantity).
Now, some words about Bourbaki. Yes, "they" uses $Q$ for rational numbers, and no, they does not use blackboard bold $\mathbb{Q}$ (at least in 1940s papers). An early occurence (maybe the earliest printed on paper) of $Q$ to denote the set of rational numbers is here at page 3 in the number 5 (7-10 December 1940) of La Tribu, the Bourbaki's internal newsletter. We read
$Q$ est ordonné [...] Topologie de $Q$ [...] Complétion de $Q$ : nombres réels
so there is no doubt that here $Q$ refers to our $\mathbb{Q}$. Clearly, we find $Q$ for rational numbers in the 1942 Algèbre (from page 29).
