the quantity of vapor contained in the unit mass of moist air which can be conveniently called the "specific humidity"

(p.327)

At that time, $q$ was not used, but instead von Bezold chose $y$ as the symbolic representation of specific humidity in his subsequent equations:

$y$ the specific humidity or the quantity of vapor in a unit mass of moist air expressed in the fractional parts of this unit.

(p.328)

The first appearance of $q$ as a symbol for specific humidity that I've found is in Chapter 5 of Physics of the Earth, Vol. 1 (p. 151), published in 1931. Here, the author uses $q$ to mean specific humidity as though it is common knowledge, using it in equations before even defining it for the reader:

This makes $q$ a readily determined and quite conservative characteristic of any unsaturated air mass.

Followed a few sentences later by:

The relative humidity $f$ together with the specific humidity $q$ and absolute humidity $\rho_{w}$ are the quantities commonly used as measures of the moisture content of the atmosphere.

(p.137)

This 1931 reference is likely not the earliest use of $q$ to mean specific humidity, but is the earliest I was able to uncover.

Thus, it would seem that the use of $q$ as a commonly accepted symbolic representation of specific humidity originated sometime between 1884 and 1931. It seems quite plausible that, given the use of the word "quantity" in defining the term originally (and in later references that I'm unable to add links for), the choice of $q$ as the symbol for specific humidity finds its source in this definition.

Additional reference: Temperature inversions in relation to frosts, McAdie, Alexander, Cambridge, 1915. (p.5) (Full text at Hathi Trust)