The family of $J$-reflection groups can be seen as a combinatorial generalisation of irreducible rank two complex reflection groups and was introduced by the author in a previous article. In this article, we define the braid groups associated to $J$-reflection groups, which coincide with the complex braid group when the $J$-reflection group is finite. We show that the isomorphism type of the braid groups only depend on the reflection isomorphism types of the corresponding $J$-reflection groups. Moreover, we show that these braid groups are always abstractly isomorphic to circular groups. At the same time, we show that the center of the braid groups is cyclic and sent onto the center of the corresponding $J$-reflection groups under the natural quotient. Finally, we exhibit two Garside structures for each braid group of $J$-reflection group. These structures generalise the classical and dual Garside structures (when defined) of rank two irreducible complex reflection groups. In particular, the dual Garside structure of $J$-reflection groups provides candidates for dual monoids associated to the irreducible complex reflection groups of rank two which do not already have one.