A connection between the quasi-biennial oscillation (QBO), the 11-yr solar cycle (SC), and the short-term convective climate oscillation, the Madden-Julian Oscillation (MJO), in boreal winter has been found in observational data, yet is generally lacking in current global climate models (GCMs). The extent to which a proposed mechanism for producing this connection 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 this mechanism, with particular issues being QBOs that are westerly-biased and weak in the lower stratosphere, weak solar stratospheric temperature and zonal wind signals, 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 by-passing 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 additional feedbacks may be needed to reproduce the full modulation.