No. The COVID-19 vaccination does not affect fertility.
No antibodies against the placenta are produced as a result of the vaccination. The vaccination stimulates the body’s immune response to the coronavirus. This immune response is aimed very specifically at the coronavirus, and does not affect fertility.
After the vaccination, the body recognises the coronavirus by its typical spike proteins, and is able to fight the virus on a targeted basis. The coronavirus’s spike protein has a slight similarity to a human protein involved in the development of the placenta. But this similarity is far too small for the body to be confused. It would be impossible for the antibodies that target the coronavirus’s spike protein to also target the body’s placenta development protein.
You can think of it something like this: two phone numbers both contain the digit 4. But they are different in every other respect. If you dial one of the numbers, you specifically reach the person with that number – and not the person with the second number. Similarly, the antibodies stimulated by the vaccination only target the spike protein of the coronavirus.