Is the Flynn effect related to the discovery of Fetal Alcohol Syndrome? Is it possible that the measured rise in average IQ in the last century is related to the discovery that drinking alcohol during pregnancy can result in birth defects, including issues with brain development?

Has any work been done on this question?
TL;DR

The story is not that clear cut. The hyposthesis is possible, however the incidence levels of Foetal Alcohol Syndrome (FAS) has risen over the years.  Speculation might say that the higher IQ averages would indicate that the average severity of FAS has decreased, but I cannot find any study to back that up.

Full answer

I would say that in order to answer this question you need to look at the framework of your question and the whole question centres on IQ (Intelligence Quotient) and Fetal Alcohol Syndome.  So let's break this up into the 2 components.

IQ (Intelligence Quotient)

The Intelligence Quotient is a total score derived from several standardised tests designed to assess human intelligence, and human intelligence involves the cognitive abilities to learn, form concepts, understand, apply logic, and reason, including the capacities to recognize patterns, comprehend ideas, plan, solve problems, make decisions, retain information, and use language to communicate. Intelligence enables humans to experience and think.

Foetal Alcohol Syndrome (FAS)

A baby exposed to alcohol in the womb may have learning difficulties such as problems with thinking, speech, maths or memory (NHS, 2017) — all needed for high intelligence — and foetal alcohol syndrome is a type of foetal alcohol spectrum disorder (FASD), the name for all the various problems that can affect children if their mother drinks alcohol in pregnancy (CDC, 2018).

CDC (2018) also states that a person with an FASD might have:


Difficulty with attention
Poor memory
Difficulty in school (especially with math)
Learning disabilities
Speech and language delays
Poor reasoning and judgment skills, and
Intellectual disability or low IQ


With all this in mind, it is possible that with lower incidences of FAS due to lower numbers of drinking pregnant mothers, comes higher average IQ numbers.

The following studies looked at this hypothesis

FAS and corresponding IQ levels


  Riikonen, R., Salonen, I., Partanen, K., &amp; Verho, S. (1999). Brain perfusion SPECT and MRI in foetal alcohol syndrome.
  
  Eleven  children  with  a  diagnosis  of  FAS  (six  boys  and  five
  girls), with a mean age of 8.6 (range 3 to 13) years, participated in the study. [...] Tests showed moderate learning disability in two children (IQ 40 and 46, respectively) (subjects 2 and 11) and mild learning disability (IQ 68 to 85) or normal intelligence (IQ >85) in the others.


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  Streissguth, A. P., Herman, C. S., &amp; Smith, D. W. (1978). Intelligence, behavior, and dysmorphogenesis in the fetal alcohol syndrome: A report on 20 patients.
  
  In a sample of 20 patients with the fetal alcohol syndrome, ages 9 months to 21 years, the average IQ was 65, with a range of 16 to 105; 60% of the patients had IQ's more than two standard deviations below the mean. The sample included patients who ranged in severity of dysmorphogenesis from mild to severe. The severity of the dysmorphic features was related to degree of mental deficiency; children with the most severe manifestations of FAS had an average IQ of 55, whereas children with lesser manifestations had an average IQ of 82. All children were growth deficient for height and/or head circumference, but in this study only height was significantly correlated with IQ.


Incidence levels of FAS


  Abel, E. L. (1995). An update on incidence of FAS: FAS is not an equal opportunity birth defect.
  
  In a previous survey (Abel &amp; Sokol, 1991), the incidence of Fetal Alcohol Syndrome (FAS) in the Western world was estimated at 0.33 cases per 1,000 live born, with a higher incidence among low SES/ African Americans than among middle-high SES/Caucasians. That eariier survey was based on 15 prospective studies of general obstetric populations appearing for prenatal care. Since that analysis, there has been almost a two-fold increase in such studies, allowing for a more up-to-date estimate of the incidence of this anomaly and a clearer perspective of at-risk populations. 
  
  The incidence of Fetal Alcohol Syndrome is now estimated at 0.97 cases per 1,000 live births in the general obstetric population and 4.3% among “heavy” drinkers. The general incidence is more than 20 times higher in the United States (1.95 per 1,000) compared to Europe and other countries (0.08 per 1,000). Within the United States, the incidence at sites characterized by low socioeconomic status, and African American or Native American background are about 10 times higher (2.29 cases per 1,000) compared to sites with a predominant middle/upper SES and Caucasian background (0.26 per 1,000). Based on racial background, the number of pregnant women in the U.S. giving birth to FAS children is 2,043 per year; if based on socioeconomic status, the number is slightly higher 2,366. Although race and SES are confounded in the U.S. studies, an examination of U.S. and European studies suggests that the major factor associated with FAS is low SES rather than racial background.


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  Habbick, B. F., Nanson, J. L., Snyder, R. E., Casey, R. E., &amp; Schulman, A. L. (1996). Foetal alcohol syndrome in Saskatchewan: unchanged incidence in a 20-year period.
  
  Despite major initiatives in public and professional education about foetal alcohol syndrome (FAS) in Saskatchewan in the last 20 years, its incidence rate has not fallen. The rate was 0.515 per 1,000 live births in 1973-1977 and 0.589 in 1988-1992.


Conclusion

The story is not that clear cut. The hyposthesis is possible, however the incidence levels of Foetal Alcohol Syndrome (FAS) has risen over the years. Speculation might say that the higher IQ averages would indicate that the average severity of FAS has decreased, but I cannot find any study to back that up.

References

Abel, E. L., &amp; Sokol, R. J. (1991). A revised conservative estimate of the incidence of FAS and its economic impact. Alcoholism: Clinical and experimental research, 15(3), 514-524. doi: 10.1111/j.1530-0277.1991.tb00553.x

Abel, E. L. (1995). An update on incidence of FAS: FAS is not an equal opportunity birth defect. Neurotoxicology and teratology, 17(4), 437-443. doi: 10.1016/0892-0362(95)00005-C

CDC (2018). Fetal alcohol spectrum disorders (FASDs) [Online]Retrieved from: https://www.cdc.gov/ncbddd/fasd/facts.html

Habbick, B., Nanson, J., Snyder, R., Casey, R., &amp; Schulman, A. (1996). Foetal Alcohol Syndrome in Saskatchewan: Unchanged Incidence in a 20-year Period. Canadian Journal of Public Health / Revue Canadienne De Sante'e Publique, 87(3), 204-207. Retrieved from: http://www.jstor.org/stable/41991432 PMID: 8771927

NHS (2017). Foetal alcohol syndrome [Online]
Retrieved from: https://www.nhs.uk/conditions/foetal-alcohol-syndrome

Riikonen, R., Salonen, I., Partanen, K., &amp; Verho, S. (1999). Brain perfusion SPECT and MRI in foetal alcohol syndrome. Developmental Medicine &amp; Child Neurology, 41(10), 652-659. doi: 10.1111/j.1469-8749.1999.tb00518.x

Streissguth, A. P., Herman, C. S., &amp; Smith, D. W. (1978). Intelligence, behavior, and dysmorphogenesis in the fetal alcohol syndrome: A report on 20 patients. The Journal of pediatrics, 92(3), 363-367. doi: 10.1016/S0022-3476(78)80420-X