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A connection between the quasi‐biennial oscillation (QBO), solar variability, and the short‐term convective climate oscillation, the Madden‐Julian oscillation (MJO), in boreal winter has been found in observational data, yet it is generally lacking in current global climate models (GCMs). A proposed mechanism is changes in tropical lower stratospheric upwelling rates and static stability caused by QBO and solar UV effects on extratropical wave forcing of the stratospheric residual meridional circulation (the Brewer‐Dobson circulation). The extent to which this mechanism, which operates only in boreal winter and enhances similar effects of the QBO‐induced meridional circulation, is simulated in a series of GCMs participating in the Coupled Model Intercomparison Project 6 (CMIP6) is investigated. The models are found to be often lacking complete representation of several elements of the mechanism, with particular issues being QBOs that are westerly biased and weak in the lower stratosphere, insufficient solar or QBO modulation of extratropical wave activity (the Holton‐Tan effect), too weak reductions in equatorial tropopause static stability in response to extratropical wave forcing, and MJOs that in some cases do not respond to these reductions. Through bypassing many of these deficiencies via data selection, it is demonstrated that effects on the MJO that resemble those found in observations (strengthening of the MJO following early winter sudden stratospheric warmings and during easterly QBO winters) can be simulated by a subset of the models. This supports operation of the proposed mechanism, and points to needed model improvements, although caveats exist and further work is needed.