This is highly unlikely, similarly to claims that eating cruciferous vegetables will lead to gassy breast-fed babies. Breast milk isn't the distilled contents of the parent's stomach. In contrast, alcohol (for example) is a problem because it affects the parent's bloodstream, not because it fills their stomach.

A) The tastes/smells of many fluids humans produce are noticeably affected by the foods they eat

B) Many with experience with milk-generating livestock will cite that the feed has a meaningful effect on the flavor of the milk.

Why would human breast milk be the exception to both of those premises?

The percentage of alcohol that makes it into breast milk, which is the mother’s blood alcohol content, is less than the percentage in a typical slice of yeast leavened bread[1]. Coma level alcohol abuse is a BAC of 0.4%, which is the around the minimum found in bread. It’s very important to remember that this intake is going through baby’s gut and liver. On the other hand drinking while pregnant is essentially putting baby on an alcohol IV and is extremely harmful.

The real danger of alcohol for breastfeeding is that it can impair the mother’s ability to properly hold the child, among other things.

Obviously I don’t advocate drinking and nursing, but if mom does have a couple glasses of wine with dinner she shouldn’t be wracked with guilt if she has to nurse.

[1] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1709087/?page=1

On the alcohol thing, I'm inclined to agree. Even if you drink enough to be legally drunk, the milk should only be 0.08% alcohol, which is considerably less than say, orange juice. This of course does not at all apply to a pregnant woman.

On the cruciferous vegetable thing, I can say that it is a real phenomenon at least for some babies in certain circumstances. One of our babies was particularly prone to gas, and we had a few horrible nights. Finally, after the worst night of crying we'd ever had, it dawned on us that my wife had eaten a salad with raw brussels sprouts and also eaten cooked brussels sprouts that night. Once she cut out the brussels sprouts and went easy on the broccoli, the problem mostly went away.

Whatever the chemical in brussels sprouts that causes gas is, evidently it is active at pretty low concentrations at least for certain babies.

It’s not entirely far fetched. Grandpa used to say that if the cows got into garlic it ruined the milk, some flavors carry over.

I live on a dairy. Yes, when our cows grazed on our cover crop which included radishes, it affected the flavor of the milk. And the color, too.

We actually had some customers tell us later that they wouldn't have kept buying our milk due to the unpleasant flavor, except that they knew it was temporary.

I'll second this; my parents had milk goats while I was young, and their diet definitely affected the milk flavor. Garlic, onions, even some weeds would do it.