Chatterjee and Garg [Proc. Amer. Math. Soc. 151 (2023), pp. 2011-2022] established closed form for a
q
q
-analogue of the Euler-Stieltjes constants. In this article, we aim to build upon their work by extending it to a
q
q
-analogue of the double zeta function. Specifically, we derive a closed form expression for
γ
0
,
0
(
q
)
\gamma _{0,0}(q)
which is a
q
q
-analogue of Euler’s constant of height
2
2
and appear as the constant term in the Laurent series expansion of a
q
q
-analogue of the double zeta function around
s
1
=
1
s_1 = 1
and
s
2
=
1
s_2=1
.
Moreover, we examine the linear independence of a set of numbers involving the constant
γ
0
′
∗
(
q
i
)
\gamma _0^{\prime *}(q^i)
, where
1
≤
i
≤
r
1 \leq i \leq r
for any integer
r
≥
1
r \geq 1
, that appears in the Laurent series expansion of a
q
q
-double zeta function. Finally, we discuss the irrationality of certain numbers involving a
2
2
-double Euler-Stieltjes constant (
γ
0
,
0
(
2
)
\gamma _{0,0}(2)
).