Not that I know of. Quota sampling (recruiting participants by gender, age, and/or other demographic characteristics in the same proportions as in the population of interest) might be less prone to bias than pure convenience sampling, but not in any know-able, systematic or quantifiable ways. So the generalizability of your results (aka, its "external validity") will always be open to question.

However, even if you cannot randomly recruit participants into the study, perhaps you can randomly assign them to the different (quasi-) experimental conditions. If so, then any "significant" differences across experimental groups would not be attributable to factors other than the experimental conditions, since all such other factors should be randomly distributed across the groups. That should reduce bias (aka, increase "internal validity") in your findings for the particular subjects recruited.

Combining quota sampling with random assignment to conditions could be a pragmatic, and defensible, approach for your research.

You can use propensity scores (assuming appropriate data are available) to create matched treatment and comparison groups.