Generally the answer is yes. The admetan package in Stata (part of the ipdmetan package) can now run Quality effects [1] and IVhet [2] meta-analyses.
Both are replacements to the random effects model that does not seem to be fit for purpose.
Example codes are given below.
For IVhet:
admetan cases1 noncases1 cases2 noncases2, re(ivhet) rr
For quality effects:
admetan cases1 noncases1 cases2 noncases2, qe(rescaled) rr
The "rescaled" variable stands for quality score rescaled between 0 and 1 by dividing the raw score by the maximum score in the list so that every meta-analysis always has a study with a rescaled score of 1.
I would recommend that users of Stata abandon random effects of all types that assume a normal distribution of "true" effects with a common variance (nuisance variable). The IVhet model does not make this assumption [3] and thus stands apart from all variance correction models as the remaining do make this assumption (e.g. hksj, bs, bt, gamma, pl, kr).
NB There is an option to use both these models with the REML estimate of tau sq - these should NOT be used as then these models are NO LONGER quality effects or IVhet models
References
1. Doi SA, Barendregt JJ, Khan S, Thalib L, Williams GM. Advances in the meta-analysis of heterogeneous clinical trials II: The quality effects model. Contemp Clin Trials. 2015 Nov;45(Pt A):123-9.
2. Doi SA, Barendregt JJ, Khan S, Thalib L, Williams GM. Advances in the meta-analysis of heterogeneous clinical trials I: The inverse variance heterogeneity model. Contemp Clin Trials. 2015 Nov;45(Pt A):130-8.
3. Doi SAR, Furuya-Kanamori L, Thalib L, Barendregt JJ. Meta-analysis in evidence-based healthcare: a paradigm shift away from random effects is overdue. Int J Evid Based Healthc. 2017 Dec;15(4):152-160.