We initiate a new, computational approach to a classical problem: certifying non-freeness of (
2
2
-generator, parabolic) Möbius subgroups of
SL
(
2
,
Q
)
\operatorname {SL}(2,\mathbb {Q})
. The main tools used are algorithms for Zariski dense groups and algorithms to compute a presentation of
SL
(
2
,
R
)
\operatorname {SL}(2, R)
for a localization
R
=
Z
[
1
b
]
R= \mathbb {Z}[\frac {1}{b}]
of
Z
\mathbb {Z}
. We prove that a Möbius group
G
≤
SL
(
2
,
R
)
G\leq \operatorname {SL}(2,R)
is not free by showing that it has finite index in
SL
(
2
,
R
)
\operatorname {SL}(2, R)
. Further information about the structure of
G
G
is obtained; for example, we compute the minimal subgroup of finite index in
SL
(
2
,
R
)
\operatorname {SL}(2,R)
containing
G
G
.