It does not appear to be significantly different from the form listed here with (in article notation) $i = {1}$, $ A = a$, and $b$, $d$ as scalars. Or is $m$ something special?
If it is possible, what is the equivalent SOC constraint? If not, can this be solved with a specialised solver, or should I defer to more general non linear optimizers?
Just rewrite the constraint as
$y= a^Tx-b$
$||y||_2 \leq \sqrt d$
and you are back to SOC. Or you can set $(0.5,d,a^Tx -b)\in Q_r$, where $Q_r$ is the rotated cone.